# 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**

### 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.

### Overlap–add method

**overlap-addoverlap–addoverlap add**

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

### Antenna (radio)

**antennaantennasradio antenna**

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.