Window Functions

Reference for common window functions used in signal processing (tapering) before spectral analysis.

All functions on this page are static methods on NDArray and return a Float64 array of shape [m].

Symmetric vs periodic

All window functions accept a $periodic flag:

• periodic = false (default): returns a symmetric window (useful for filter design)
• periodic = true: returns a periodic window (often used for FFT / spectral analysis)

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bartlett()

public static function bartlett(int $m, bool $periodic = false): NDArray

Bartlett (triangular) window with endpoints at zero for m > 1. Often used to taper a signal without introducing strong discontinuities at the boundaries.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Notes

For m > 1, the Bartlett window is a piecewise-linear triangle that reaches 1.0 in the center and tapers to 0.0 at both ends.

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triang()

public static function triang(int $m, bool $periodic = false): NDArray

Triangular window. Similar to Bartlett, but does not necessarily force the endpoints to zero.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

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blackman()

public static function blackman(int $m, bool $periodic = false): NDArray

Blackman window, formed using the first three terms of a cosine series. Designed for low spectral leakage (better sidelobe suppression than Hann/Hamming at the expense of a wider main lobe).

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Formula

For m > 1 and 0 ≤ n ≤ m-1:

\[ w(n) = 0.42 - 0.5\cos\left(\frac{2\pi n}{m-1}\right) + 0.08\cos\left(\frac{4\pi n}{m-1}\right) \]

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hamming()

public static function hamming(int $m, bool $periodic = false): NDArray

Hamming window, a raised cosine with non-zero endpoints. Commonly used to reduce the nearest sidelobe levels compared to the Hann window.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Formula

For m > 1 and 0 ≤ n ≤ m-1:

\[ w(n) = 0.54 - 0.46\cos\left(\frac{2\pi n}{m-1}\right) \]

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hanning() / hann()

public static function hanning(int $m, bool $periodic = false): NDArray
public static function hann(int $m, bool $periodic = false): NDArray

Hanning (Hann) window, a raised cosine with endpoints at zero for m > 1. Useful for smoothing discontinuities at the beginning and end of a signal segment.

hann() is an alias of hanning().

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Formula

For m > 1 and 0 ≤ n ≤ m-1:

\[ w(n) = 0.5 - 0.5\cos\left(\frac{2\pi n}{m-1}\right) \]

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bohman()

public static function bohman(int $m, bool $periodic = false): NDArray

Bohman window. A smooth window with good sidelobe behavior derived from convolving two cosine-tapered windows.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Formula

Let (x = \left|\frac{2n}{m-1} - 1\right|). For m > 1:

\[ w(n) = (1-x)\cos(\pi x) + \frac{\sin(\pi x)}{\pi} \]

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boxcar()

public static function boxcar(int $m, bool $periodic = false): NDArray

Boxcar (rectangular) window. This is the “no windowing” case (all ones).

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

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lanczos()

public static function lanczos(int $m, bool $periodic = false): NDArray

Lanczos window. Defined using a normalized sinc, it is commonly used in interpolation and resampling contexts.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].

Formula

Let (x = \frac{2n}{m-1} - 1\). For m > 1:

\[ w(n) = \mathrm{sinc}(x), \quad \mathrm{sinc}(0)=1,\; \mathrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x} \]

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kaiser()

public static function kaiser(int $m, float $beta, bool $periodic = false): NDArray

Kaiser window, parameterized by beta. Larger beta increases sidelobe attenuation (less leakage) at the cost of a wider main lobe.

Parameters

Parameter	Type	Description
$m	int	Number of points in the output window. If 0, an empty array is returned. If 1, returns [1.0].
$beta	float	Shape parameter controlling the trade-off between main-lobe width and sidelobe level.
$periodic	bool	If true, returns the periodic variant. Default: false.

Returns

• NDArray - Float64 window of shape [m].