# Window function

window functionswindowingwindowedwindow windowingBlackman-Harris Filteranalysis windowsBartlettBlackman–Harris filterboxcar window
In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.wikipedia
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### Discrete Fourier transform

DFTcircular convolution theoremFourier transform
When the input waveform is time-sampled, instead of continuous, the analysis is usually done by applying a window function and then a discrete Fourier transform (DFT).
In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function ).

### Beamforming

beam formingbeamformerAntenna beamforming
Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design.
Different weighting patterns (e.g., Dolph-Chebyshev) can be used to achieve the desired sensitivity patterns.

### Kernel (statistics)

kernelkernelskernel estimation
In the field of Bayesian analysis and curve fitting, this is often referred to as the kernel.
The term kernel is used in statistical analysis to refer to a window function.

### Spectral leakage

leakagemain lobe
But that method also changes the frequency content of the signal by an effect called spectral leakage.
But the term 'leakage' usually refers to the effect of windowing, which is the product of s(t) with a different kind of function, the window function.

### Discrete-time Fourier transform

Sampling the DTFT discrete-time Fourier transformconvolution theorem
But the DFT provides only a sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum.
For instance, a long sequence might be truncated by a window function of length resulting in two cases worthy of special mention:

### Finite impulse response

FIRFIR filterFinite Impulse Response (FIR)
Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna 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.
In the window design method, one first designs an ideal IIR filter and then truncates the infinite impulse response by multiplying it with a finite length window function.

### Fredric J. Harris

Harris
Both sinusoids suffer less SNR loss under the Hann window than under the Blackman–Harris window.
He is also the co-inventor of the Blackman-Harris Filter.

### Richard Hamming

Hamming Hamming, RichardHamming, Richard
Setting a_0 to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by Richard W. Hamming.
His contributions include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), and the Hamming distance.

### Spectral density

frequency spectrumpower spectrumspectrum
Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design. A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect.
Window function

### Julius von Hann

Julius Ferdinand von HannJulius Hann
named after Julius von Hann, and sometimes referred to as Hanning, presumably due to its linguistic and formulaic similarities to the Hamming window.
In signal processing, the Hann window is a window function, called the Hann function, named after him by R. B. Blackman and John Tukey in "Particular Pairs of Windows," published in "The Measurement of Power Spectra, From the Point of View of Communications Engineering", New York: Dover, 1959, pp. 98–99.

### Multitaper

Multitaper method
The DPSS (discrete prolate spheroidal sequence) or Slepian window maximizes the energy concentration in the main lobe, and is used in multitaper spectral analysis, which averages out noise in the spectrum and reduces information loss at the edges of the window.
Each data taper is multiplied element-wise by the signal to provide a windowed trial from which one estimates the power at each component frequency.

### Modified discrete cosine transform

MDCTMDCT-basedMDCTs
See Welch method of power spectral analysis and the modified discrete cosine transform.
In typical signal-compression applications, the transform properties are further improved by using a window function w n (n = 0, ..., 2N−1) that is multiplied with x n and y n in the MDCT and IMDCT formulas, above, in order to avoid discontinuities at the n = 0 and 2N boundaries by making the function go smoothly to zero at those points.

### Signal processing

signal analysissignalsignal processor
In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

### Statistics

statisticalstatistical analysisstatistician
In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

### Function (mathematics)

functionfunctionsmathematical function
In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

### Interval (mathematics)

intervalopen intervalclosed interval
In signal processing and statistics, a window function (also known as an apodization function or tapering function ) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

### Square-integrable function

square-integrablesquare integrableL'' 2
A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.

Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design.

Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design.

### Fourier transform

Fouriercontinuous Fourier transformuncertainty principle
The Fourier transform of the function cos ωt is zero, except at frequency ±ω.

### Dynamic range

DRdynamicdynamic and tonal range
This characteristic is sometimes described as low dynamic range.

### Noise floor

floornoise-floor
A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect.

### Signal-to-noise ratio

signal to noise ratioSNRsignal-to-noise
Effectively, the signal to noise ratio (SNR) is improved by distributing the noise uniformly, while concentrating most of the sinusoid's energy around one frequency.

### Ralph Beebe Blackman

Blackman
Both sinusoids suffer less SNR loss under the Hann window than under the Blackman–Harris window.

### Digital filter

filterdigitaldigital filters
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.