In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. Several ''
window functions
A window is an opening in a wall, door, roof, or vehicle that allows the exchange of light and may also allow the passage of sound and sometimes air. Modern windows are usually glazed or covered in some other transparent or translucent mater ...
'' can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multiplying the given sampled signal by the window function.
Conventions
*
is a zero-phase function (symmetrical about
),
continuous for
where
is a positive integer (even or odd).
* The sequence
is ''symmetric'', of length
*
is ''DFT-symmetric'', of length
* The parameter B displayed on each spectral plot is the function's noise equivalent bandwidth metric, in units of ''DFT bins''.
The sparse sampling of a DTFT (such as the DFTs in Fig 1) only reveals the leakage into the DFT bins from a sinusoid whose frequency is also an integer DFT bin. The unseen sidelobes reveal the leakage to expect from sinusoids at other frequencies. Therefore, when choosing a window function, it is usually important to sample the DTFT more densely (as we do throughout this section) and choose a window that suppresses the sidelobes to an acceptable level.
Rectangular window
![Window_function_and_frequency_response_-_Rectangular](https://upload.wikimedia.org/wikipedia/commons/6/6a/Window_function_and_frequency_response_-_Rectangular.svg)
The rectangular window (sometimes known as the
boxcar
A boxcar is the North American ( AAR) term for a railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simplest freight car design, is considered one of the most versatile since it can carry most ...
or
Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
window) is the simplest window, equivalent to replacing all but ''N'' values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off:
:
Other windows are designed to moderate these sudden changes, which reduces scalloping loss and improves dynamic range, as described above ().
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).
Alternative definitions sample the appropriate normalized
''B''-spline basis functions instead of convolving discrete-time windows. A ''k''
th-order ''B''-spline basis function is a piece-wise polynomial function of degree ''k''−1 that is obtained by ''k''-fold self-convolution of the
rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{rl ...
.
Triangular window
Triangular windows are given by:
:
where ''L'' can be ''N'',
''N'' + 1,
or ''N'' + 2.
The first one is also known as
Bartlett window or
Fejér window. All three definitions converge at large ''N''.
The triangular window is the 2nd order ''B''-spline window. The ''L'' = ''N'' form can be seen as the convolution of two ''N''/2-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.
Parzen window
![Window_function_and_frequency_response_-_Parzen](https://upload.wikimedia.org/wikipedia/commons/4/48/Window_function_and_frequency_response_-_Parzen.svg)
Defining , the Parzen window, also known as the de la Vallée Poussin window,
is the 4th order ''B''-spline window given by:
:
:
Other polynomial windows
Welch window
The Welch window consists of a single
parabolic section:
:
The defining
quadratic polynomial
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
reaches a value of zero at the samples just outside the span of the window.
Sine window
![Window_function_and_frequency_response_-_Cosine](https://upload.wikimedia.org/wikipedia/commons/e/e5/Window_function_and_frequency_response_-_Cosine.svg)
:
The corresponding
function is a cosine without the /2 phase offset. So the ''sine window''
is sometimes also called ''cosine window''.
As it represents half a cycle of a sinusoidal function, it is also known variably as ''half-sine window''
or ''half-cosine window''.
The
autocorrelation of a sine window produces a function known as the Bohman window.
Power-of-sine/cosine windows
These window functions have the form:
:
The
rectangular window (), the
sine window (), and the
Hann window
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital sign ...
() are members of this family.
For even-integer values of these functions can also be expressed in cosine-sum form:
:
:
Cosine-sum windows
This family is also known as
generalized cosine windows'.
In most cases, including the examples below, all coefficients ''a''
''k'' ≥ 0. These windows have only 2''K'' + 1 non-zero ''N''-point DFT coefficients.
Hann and Hamming windows
The customary cosine-sum windows for case ''K'' = 1 have the form:
:
which is easily (and often) confused with its zero-phase version:
:
Setting
produces a Hann window:
:
named after
Julius von Hann
Julius Ferdinand von Hann (23 March 1839 in Wartberg ob der Aist near Linz – 1 October 1921 in Vienna) was an Austrian meteorologist. He is seen as a father of modern meteorology.
Biography
He was educated at the gymnasium of Kremsmünster a ...
, and sometimes erroneously referred to as ''Hanning'', presumably due to its linguistic and formulaic similarities to the Hamming window. It is also known as raised cosine, because the zero-phase version,
is one lobe of an elevated cosine function.
This function is a member of both the
cosine-sum and
power-of-sine families. Unlike the
Hamming window
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several '' window functions'' can be defined, based on ...
, the end points of the Hann window just touch zero. The resulting
side-lobes roll off at about 18 dB per octave.
Setting
to approximately 0.54, or more precisely 25/46, produces the Hamming window, proposed by
Richard W. Hamming. That choice places a zero-crossing at frequency 5/(''N'' − 1), which cancels the first sidelobe of the Hann window, giving it a height of about one-fifth that of the Hann window.
The Hamming window is often called the Hamming blip when used for
pulse shaping In electronics and telecommunications, pulse shaping is the process of changing the waveform of transmitted pulses to optimize the signal for its intended purpose or the communication channel. This is often done by limiting the bandwidth of the tra ...
.
Approximation of the coefficients to two decimal places substantially lowers the level of sidelobes,
to a nearly equiripple condition.
In the equiripple sense, the optimal values for the coefficients are a
0 = 0.53836 and a
1 = 0.46164.
Blackman window
Blackman windows are defined as:
:
:
By common convention, the unqualified term ''Blackman window'' refers to Blackman's "not very serious proposal" of (''a''
0 = 0.42, ''a''
1 = 0.5, ''a''
2 = 0.08), which closely approximates the exact Blackman,
with ''a''
0 = 7938/18608 ≈ 0.42659, ''a''
1 = 9240/18608 ≈ 0.49656, and ''a''
2 = 1430/18608 ≈ 0.076849.
These exact values place zeros at the third and fourth sidelobes,
but result in a discontinuity at the edges and a 6 dB/oct fall-off. The truncated coefficients do not null the sidelobes as well, but have an improved 18 dB/oct fall-off.
Nuttall window, continuous first derivative
The continuous form of the Nuttall window,
and its first
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
are continuous everywhere, like the
Hann function
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital s ...
. That is, the function goes to 0 at unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows. The Blackman window () is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not.
:
:
Blackman–Nuttall window
![Window_function_and_frequency_response_-_Blackman-Nuttall](https://upload.wikimedia.org/wikipedia/commons/b/bd/Window_function_and_frequency_response_-_Blackman-Nuttall.svg)
:
:
Blackman–Harris window
![Window_function_and_frequency_response_-_Blackman-Harris](https://upload.wikimedia.org/wikipedia/commons/0/0a/Window_function_and_frequency_response_-_Blackman-Harris.svg)
A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels
:
:
Flat top window
![Window_function_and_frequency_response_-_flat_top](https://upload.wikimedia.org/wikipedia/commons/2/27/Window_function_and_frequency_response_-_flat_top.svg)
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.
Drawbacks of the broad bandwidth are poor frequency resolution and high .
Flat top windows can be designed using low-pass filter design methods,
or they may be of the usual
cosine-sum variety:
:
Th
Matlab varianthas these coefficients:
:
Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.
Rife–Vincent windows
Rife–Vincent windows
are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied to , reflect that custom.
Class I, Order 1 (''K'' = 1):
Functionally equivalent to the
Hann window
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital sign ...
.
Class I, Order 2 (''K'' = 2):
Class I is defined by minimizing the high-order sidelobe amplitude. Coefficients for orders up to K=4 are tabulated.
Class II minimizes the main-lobe width for a given maximum side-lobe.
Class III is a compromise for which order ''K'' = 2 resembles the .
Adjustable windows
Gaussian window
The Fourier transform of a
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either be truncated at the ends of the window, or itself windowed with another zero-ended window.
Since the log of a Gaussian produces a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, this can be used for nearly exact quadratic interpolation in
frequency estimation
In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signa ...
.
:
:
The standard deviation of the Gaussian function is ''σ'' · ''N''/2 sampling periods.
Confined Gaussian window
The confined Gaussian window yields the smallest possible root mean square frequency width for a given temporal width .
These windows optimize the RMS time-frequency bandwidth products. They are computed as the minimum eigenvectors of a parameter-dependent matrix. The confined Gaussian window family contains the and the in the limiting cases of large and small , respectively.
Approximate confined Gaussian window
Defining , a
confined Gaussian window of temporal width is well approximated by:
:
where
is a Gaussian function:
::
The standard deviation of the approximate window is
asymptotically equal
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...
(i.e. large values of ) to for .
Generalized normal window
A more generalized version of the Gaussian window is the generalized normal window.
Retaining the notation from the
Gaussian window
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several '' window functions'' can be defined, based on ...
above, we can represent this window as
:
for any even
. At
, this is a Gaussian window and as
approaches
, this approximates to a rectangular window. The
Fourier transform of this window does not exist in a closed form for a general
. However, it demonstrates the other benefits of being smooth, adjustable bandwidth. Like the , this window naturally offers a "flat top" to control the amplitude attenuation of a time-series (on which we don't have a control with Gaussian window). In essence, it offers a good (controllable) compromise, in terms of spectral leakage, frequency resolution and amplitude attenuation, between the Gaussian window and the rectangular window.
See also
for a study on
time-frequency representation of this window (or function).
Tukey window
The Tukey window, also known as the ''cosine-tapered window'', can be regarded as a cosine lobe of width (spanning observations) that is convolved with a rectangular window of width .
:
At it becomes rectangular, and at it becomes a Hann window.
Planck-taper window
The so-called "Planck-taper" window is a
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
that has been widely used
in the theory of
partitions of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
in
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
. It is
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
(a
function) everywhere, but is exactly zero outside of a compact region, exactly one over an interval within that region, and varies smoothly and monotonically between those limits. Its use as a window function in signal processing was first suggested in the context of
gravitational-wave astronomy
Gravitational-wave astronomy is an emerging branch of observational astronomy which aims to use gravitational waves (minute distortions of spacetime predicted by Albert Einstein's theory of general relativity) to collect observational data about ...
, inspired by the
Planck distribution.
It is defined as a
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
function:
:
The amount of tapering is controlled by the parameter ''ε'', with smaller values giving sharper transitions.
DPSS or Slepian window
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.
The main lobe ends at a frequency bin given by the parameter ''α''.
The Kaiser windows below are created by a simple approximation to the DPSS windows:
Kaiser window
The Kaiser, or Kaiser–Bessel, window is a simple approximation of the
DPSS window using
Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
, discovered by
James Kaiser
James Frederick Kaiser (Dec. 10, 1929 – Feb. 13, 2020) was an American electrical engineer noted for his contributions in signal processing. He was an IEEE Fellow and received many honors and awards, including the IEEE Centennial Medal, the I ...
.
:
:
where
is the zero-th order modified Bessel function of the first kind. Variable parameter
determines the tradeoff between main lobe width and side lobe levels of the spectral leakage pattern. The main lobe width, in between the nulls, is given by
in units of DFT bins,
and a typical value of
is 3.
Dolph–Chebyshev window
Minimizes the
Chebyshev norm of the side-lobes for a given main lobe width.
The zero-phase Dolph–Chebyshev window function