## Window Functions Reference

This page is a companion reference to the posts Windows of Opportunity and Windows of Opportunity – ENBW. Since putting all of the content into that post also would have made it way too long (instead of just regular too long), the discussion in it was limited to three example window functions (also called apodization functions). This page is intended to serve solely as a reference for comparison of a number of other often-seen window functions. For the meaning of the various terms, see the original blog post.

I have compressed most of the information here into a handy 3-page PDF reference. Download the Window Functions Cheat Sheet.

The visual overviews contain some of the more important properties of each window function, such as window shape and total energy, instrument function shape, main lobe height and fraction of total energy $E$ contained therein ($E_0 / E$), suppression of highest side lobe, and maximum scalloping.

Where possible, the apodization function is given. Since the window function usually itself is just a segment of an infinitely extended function, it must be bounded. This can be written as

$$w(t) = \begin{cases} A(t) & -\frac{1}{2} \le t \lt \frac{1}{2} \\ 0 & \vphantom{\frac{T}{2}}\mathrm{else} \end{cases}$$

where $A(t)$ describes the actual shape of the window and is given below.

The apodization functions are normalized such that the window width $T=1$. The instrument function corresponding to the un-normalized apodization function $A(t / T)$ is then, according to Fourier transform rules, $T \cdot I(f \cdot T)$.

The frequency axis is divided into DFT bins which are separated by $\Delta f = T^{-1}$. For the normalized functions, $\Delta f = 1$. Aliasing effects due to finite time-resolution sampling, which make the instrument function dependent on the number of time samples within the window, are not considered.

#### Box / Rectangular Window

The box / rectangular window.

$$A_\mathrm{rect}(t) = 1$$

$$I_\mathrm{rect}(f) = \mathrm{sinc} \bigl(\pi f \bigr)$$

 window energy 0dB main lobe height 0dB equivalent noise BW 1$\Delta f$ (0dB) main lobe FWHM 0.884$\Delta f$ first zero 1$\Delta f$ maximum scalloping 3.92dB highest side lobe$^1$ -13.3dB at 1.43$\Delta f$ side lobe slope 20dB/decade notes This is the window when no window is used, i.e. when the segment is simply cut out of the time signal. It has the best resolution (narrowest main lobe), but highest spectral leakage and scalloping, and lowest dynamic range. Used for detecting strong signals that are spectrally close. see also rectangular window [J. O. Smith III]

#### Hann Window

The Hann window.

$$A_\mathrm{Hann}(t) = \cos^2 \bigl(\pi t \bigr)$$

$$I_\mathrm{Hann}(f) = \frac{1}{2} \frac{\mathrm{sinc} \bigl(\pi f \bigr)}{1-f^2}$$

 window energy -4.26dB main lobe height -6.02dB equivalent noise BW 1.50$\Delta f$ (1.76dB) main lobe FWHM 1.44$\Delta f$ first zero 2$\Delta f$ maximum scalloping 1.42dB highest side lobe -31.5dB at 2.36$\Delta f$ side lobe slope 60dB/decade notes Also called the raised cosine or (wrongly) Hanning window, the Hann window is quite universal due to its balance between good resolution and dynamic range. The first side lobe is still somewhat high, but their slope is steep. The power-of-cosine window family is a generalization of the Hann window with arbitrarily steep side lobe slope. see also Hann window [J. O. Smith III]Power-of-Cosine window family [J. O. Smith III]

#### Flat-top Window

The flat-top window.

$$\begin{gather} A_\mathrm{flat}(t) = \sum_{n=0}^4 a_{n} \cos \bigl(2 n \pi t \bigr)\\ \text{with} \vphantom{\frac{1}{2}} \quad a_0 = 0.21557895 \quad a_1 = 0.41663158 \quad a_2 = 0.277263158\\ \quad a_3 = 0.083578947 \quad a_4 = 0.006947368 \end{gather}$$

 window energy -7.56dB main lobe height -13.3dB equivalent noise BW 3.75$\Delta f$ (5.74dB) main lobe FWHM 3.72$\Delta f$ first zero 5$\Delta f$ maximum scalloping 0.00978dB highest side lobe -93.6dB at 7.47$\Delta f$ side lobe slope 20dB/decade$^2$ notes The flat-top window has very little scalloping and high dynamic range, but poor resolution, and is thus used to accurately determine the height of single spectral peaks or signals with some separation between peaks. Good for high sampling rates where $\Delta f$ can be very small. see also

#### Hamming Window

The Hamming window.

$$A_\mathrm{Hamming}(t) = \frac{27}{50} + \frac{23}{50} \cos \bigl(2\pi t\bigr)$$

$$I_\mathrm{Hamming}(f) = \Bigl( \frac{27}{50} – \frac{4}{50}f^2 \Bigr)\frac{\mathrm{sinc} \bigl(\pi f \bigr)}{1-f^2}$$

 window energy -4.01dB main lobe height -5.35dB equivalent noise BW 1.36$\Delta f$ (1.34dB) main lobe FWHM 1.30$\Delta f$ first zero 2$\Delta f$ maximum scalloping 1.75dB highest side lobe -42.7dB at 4.50$\Delta f$ side lobe slope 20dB/decade notes With a main lobe still narrower than the Hann window, the Hamming window was designed to minimize the height of the first sidelobe. The side lobe slope is significantly smaller than that of the Hann window. It has a medium dynamic range. The coefficients of the apodization function may be slightly different depending on the author. see also Hamming window [J. O. Smith III]

#### Blackman-Harris Window

The Blackman-Harris windows.

$$\begin{gather} A_\mathrm{BH}(t) = \sum_{n=0}^3 a_{n} \cos \bigl(2 n \pi t \bigr)\\ \text{3-term:} \vphantom{\frac{1}{2}} \; a_0 = 0.21557895 \quad a_1 = 0.4973406 \quad a_2 = 0.0782793 \quad a_3 = 0\\ \text{4-term:} \quad a_0 = 0.35875 \; a_1 = 0.48829 \quad a_2 = 0.14128 \quad a_3 = 0.01168 \end{gather}$$

 window energy -5.13dB (3-term) / -5.88dB (4-term) main lobe height -7.44dB / -8.90dB equivalent noise BW 1.70$\Delta f$ (2.31dB) / 2.00$\Delta f$ (3.02dB) main lobe FWHM 1.62$\Delta f$ / 1.90$\Delta f$ first zero 3$\Delta f$ / 4$\Delta f$ maximum scalloping 1.14dB / 0.826dB highest side lobe -71.5dB at 3.64$\Delta f$ / -92.0dB at 4.52$\Delta f$ side lobe slope 20dB/decade / 20dB/decade$^2$ notes The Blackman-Harris windows come mainly in two flavors: a three-term and a four-term sum of cosines. Their coefficients are designed to minimize side lobe levels. see also Blackman-Harris window [Wikipedia]Blackman-Harris window family [J. O. Smith III]

#### Gaussian Window

Some Gaussian windows.

$$A_\mathrm{Gauss}(t) =\exp \biggl(-\frac{t^2}{2 \sigma^2}\biggr)$$

$$I_\mathrm{Gauss}(f) = \sqrt{\frac{\pi}{2}}\sigma \exp \bigl(-2 \pi^2 \sigma^2 f^2 \bigr) \Biggl[ \mathrm{erf} \biggl( \frac{\frac{1}{2}-2 i f \pi \sigma^2}{\sqrt{2} \sigma} \biggr) + \mathrm{erf} \biggl( \frac{\frac{1}{2}+2 i f \pi \sigma^2}{\sqrt{2} \sigma} \biggr) \Biggr]$$

 window energy -4.51dB ($\sigma = 0.20$) / -3.56dB ($\sigma = 0.25$) / -2.82 ($\sigma = 0.30$) main lobe height -8.90dB / -4.46dB / -3.35dB equivalent noise BW 2.75$\Delta f$ (4.39dB) / 1.23$\Delta f$ (0.90dB) / 1.13$\Delta f$ (0.53dB) main lobe FWHM 1.89$\Delta f$ / 1.18$\Delta f$ / 1.08$\Delta f$ first zero 3.20$\Delta f$ / 1.67$\Delta f$ / 1.38$\Delta f$ maximum scalloping 1.58dB / 2.13dB / 2.54dB highest side lobe -43.3dB at 3.65$\Delta f$ / -31.9dB at 2.62$\Delta f$ / -25.0dB at 1.70$\Delta f$ side lobe slope 20dB/decade notes Better results may be obtained when the Gaussian window is truncated (multiplied) with another window instead of the box function. see also

#### DPSS / Slepian Window

Some DPSS windows.

no closed-form apodization function available

 window energy -2.61dB ($\alpha = 1$) / -4.36dB ($\alpha = 2$) / -5.30 ($\alpha = 3$) main lobe height -3.09dB / -6.02dB / -7.78dB equivalent noise BW 1.12$\Delta f$ (0.48dB) / 1.47$\Delta f$ (1.66dB) / 1.77$\Delta f$ (2.48dB) main lobe FWHM 1.07$\Delta f$ / 1.40$\Delta f$ / 1.68$\Delta f$ first zero 1.35$\Delta f$ / 2.17$\Delta f$ / 3.11$\Delta f$ maximum scalloping 2.59dB / 1.51dB / 1.05dB highest side lobe -22.9dB at 1.69$\Delta f$ / -44.8dB at 2.39$\Delta f$ / -69.7dB at 3.26$\Delta f$ side lobe slope 20dB/decade notes The discrete prolate spheroidal sequence window is optimized to have maximum energy in its main lobe for a given time-bandwidth product. The (real-valued) parameter $\alpha$ corresponds to half this time-bandwidth product and also gives the largest DFT bin number that is still inside the main lobe. The window can be obtained with the MATLAB DPSS function. The Kaiser window is an approximation to the DPSS window using Bessel functions. The continuous equivalent, the Slepian window, is described by the first prolate spheroidal wave function. see also Slepian or DPSS window [J. O. Smith III]Kaiser window [J. O. Smith III]Kaiser window [Wikipedia]

1 Side lobe level is relative to main lobe height.

2 Side lobe decrease with this slope starts at about 20$\Delta f$.