Window function

Window functions are used in spectral analysis/modification/resynthesis, as well as beamforming and antenna design. (DTFT) also exhibits the same leakage pattern (rows 3 and 4). But when the DTFT is only sparsely sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue DTFT, those samples are the outputs of the discrete Fourier transform (DFT). The red DTFT has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed. Spectral analysis The Fourier transform of the function is zero, except at frequency ±ω. However, many other functions and waveforms do not have convenient closed-form transforms. Alternatively, one might be interested in their spectral content only during a certain time period. In either case, the Fourier transform (or a similar transform) can be applied on one or more finite intervals of the waveform. In general, the transform is applied to the product of the waveform and a window function. Any window (including rectangular) affects the spectral estimate computed by this method. Filter design Windows are sometimes used in the design of digital filters, in particular to convert an "ideal" impulse response of infinite duration, such as a sinc function, to a finite impulse response (FIR) filter design. That is called the window method. Statistics and curve fitting Window functions are sometimes used in the field of statistical analysis to restrict the set of data being analyzed to a range near a given point, with a weighting factor that diminishes the effect of points farther away from the portion of the curve being fit. In the field of Bayesian analysis and curve fitting, this is often referred to as the kernel. Rectangular window applications Analysis of transients When analyzing a transient signal in modal analysis, such as an impulse, a shock response, a sine burst, a chirp burst, or noise burst, where the energy vs time distribution is extremely uneven, the rectangular window may be most appropriate. For instance, when most of the energy is located at the beginning of the recording, a non-rectangular window attenuates most of the energy, degrading the signal-to-noise ratio. Harmonic analysis One might wish to measure the harmonic content of a musical note from a particular instrument or the harmonic distortion of an amplifier at a given frequency. Referring again to Figure 2, we can observe that there is no leakage at a discrete set of harmonically-related frequencies sampled by the discrete Fourier transform (DFT). (The spectral nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.) This property is unique to the rectangular window, and it must be appropriately configured for the signal frequency, as described above. == Overlapping windows ==

When the length of a data set to be transformed is larger than necessary to provide the desired frequency resolution, a common practice is to subdivide it into smaller sets and window them individually. To mitigate the "loss" at the edges of the window, the individual sets may overlap in time. See Welch method of power spectral analysis and the modified discrete cosine transform. == Two-dimensional windows ==

Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform. They can be constructed from one-dimensional windows in either of two forms. The separable form, W(m,n)=w(m)w(n) is trivial to compute. The radial form, W(m,n)=w(r), which involves the radius r=\sqrt{(m-M/2)^2+(n-N/2)^2}, is isotropic, independent on the orientation of the coordinate axes. Only the Gaussian function is both separable and isotropic. The separable forms of all other window functions have corners that depend on the choice of the coordinate axes. The isotropy/anisotropy of a two-dimensional window function is shared by its two-dimensional Fourier transform. The difference between the separable and radial forms is akin to the result of diffraction from rectangular vs. circular apertures, which can be visualized in terms of the product of two sinc functions vs. an Airy function, respectively. == Examples of window functions ==

Conventions: • w_0(x) is a zero-phase function (symmetrical about x=0),) is the simplest window, equivalent to replacing all but N consecutive values of a data sequence by zeros, making the waveform suddenly turn on and off: :w[n] = 1. Other windows are designed to moderate these sudden changes, to reduce scalloping loss and improve dynamic range (described in ). The rectangular window is the 1st-order B-spline window as well as the 0th-power power-of-sine window. The rectangular window provides the minimum mean square error estimate of the Discrete-time Fourier transform, at the cost of other issues discussed. B-spline windows B-spline windows can be obtained as k-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the (k = 2) and the (k = 4). {{Equation box 1 In most cases, including the examples below, all coefficients ak ≥ 0. These windows have only 2K + 1 non-zero N-point DFT coefficients. Blackman window Blackman windows are defined as :w[n] = a_0 - a_1 \cos \left ( \frac{2 \pi n}{N} \right) + a_2 \cos \left ( \frac{4 \pi n}{N} \right), :a_0=\frac{1-\alpha}{2};\quad a_1=\frac{1}{2};\quad a_2=\frac{\alpha}{2}. By common convention, the unqualified term Blackman window refers to Blackman's "not very serious proposal" of (a0 = 0.42, a1 = 0.5, a2 = 0.08), which closely approximates the exact Blackman, Flat top window A flat top window is a partially negative-valued window that has minimal scalloping loss in the frequency domain. That property is desirable for the measurement of amplitudes of sinusoidal frequency components. == See also ==