Window function

For the term used in SQL statements, see Window function (SQL)

In signal processing, 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. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or waveform/data-sequence is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap; the "view through the window". Applications of window functions include spectral analysis, filter design, and beamforming. In typical applications, the window functions used are non-negative smooth "bell-shaped" curves, though rectangle, triangle, and other functions can be used.

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, that is, that the function goes sufficiently rapidly toward zero.

Applications

Applications of window functions include spectral analysis and the design of finite impulse response filters.

Spectral analysis

The Fourier transform of the function cos ωt 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 something similar) 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.

Figure 1: Zoomed view of spectral leakage

Windowing

Windowing of a simple waveform like cos ωt causes its Fourier transform to develop non-zero values (commonly called spectral leakage) at frequencies other than ω. The leakage tends to be worst (highest) near ω and least at frequencies farthest from ω.

If the waveform under analysis comprises two sinusoids of different frequencies, leakage can interfere with the ability to distinguish them spectrally. If their frequencies are dissimilar and one component is weaker, then leakage from the larger component can obscure the weaker one’s presence. But if the frequencies are similar, leakage can render them unresolvable even when the sinusoids are of equal strength.

The rectangular window has excellent resolution characteristics for sinusoids of comparable strength, but it is a poor choice for sinusoids of disparate amplitudes. This characteristic is sometimes described as low-dynamic-range.

At the other extreme of dynamic range are the windows with the poorest resolution. These high-dynamic-range low-resolution windows are also poorest in terms of sensitivity; this is, if the input waveform contains random noise close to the frequency of a sinusoid, the response to noise, compared to the sinusoid, will be higher than with a higher-resolution window. In other words, the ability to find weak sinusoids amidst the noise is diminished by a high-dynamic-range window. High-dynamic-range windows are probably most often justified in wideband applications, where the spectrum being analyzed is expected to contain many different components of various amplitudes.

In between the extremes are moderate windows, such as Hamming and Hann. They are commonly used in narrowband applications, such as the spectrum of a telephone channel. In summary, spectral analysis involves a tradeoff between resolving comparable strength components with similar frequencies and resolving disparate strength components with dissimilar frequencies. That tradeoff occurs when the window function is chosen.

Discrete-time signals

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). But the DFT provides only a coarse sampling of the actual DTFT spectrum. Figure 1 shows a portion of the DTFT for a rectangularly windowed sinusoid. The actual frequency of the sinusoid is indicated as "0" on the horizontal axis. Everything else is leakage, exaggerated by the use of a logarithmic presentation. The unit of frequency is "DFT bins"; that is, the integer values on the frequency axis correspond to the frequencies sampled by the DFT. So the figure depicts a case where the actual frequency of the sinusoid happens to coincide with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. When it misses the maximum value by some amount (up to 1/2 bin), the measurement error is referred to as scalloping loss (inspired by the shape of the peak). But the most interesting thing about this case is that all the other samples coincide with nulls in the true spectrum. (The nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.) So in this case, the DFT creates the illusion of no leakage. Despite the unlikely conditions of this example, it is a common misconception that visible leakage is some sort of artifact of the DFT. But since any window function causes leakage, its apparent absence (in this contrived example) is actually the DFT artifact.

This figure compares the processing losses of three window functions for sinusoidal inputs, with both minimum and maximum scalloping loss.

Noise bandwidth

The concepts of resolution and dynamic range tend to be somewhat subjective, depending on what the user is actually trying to do. But they also tend to be highly correlated with the total leakage, which is quantifiable. It is usually expressed as an equivalent bandwidth, B. Think of it as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B. The more leakage, the greater the bandwidth. It is sometimes called noise equivalent bandwidth or equivalent noise bandwidth, because it is proportional to the average power that will be registered by each DFT bin when the input signal contains a random noise component (or is just random noise). A graph of the power spectrum, averaged over time, typically reveals a flat noise floor, caused by this effect. The height of the noise floor is proportional to B. So two different window functions can produce different noise floors.

Processing gain and losses

In signal processing, operations are chosen to improve some aspect of quality of a signal by exploiting the differences between the signal and the corrupting influences. When the signal is a sinusoid corrupted by additive random noise, spectral analysis distributes the signal and noise components differently, often making it easier to detect the signal's presence or measure certain characteristics, such as amplitude and frequency. 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. Processing gain is a term often used to describe an SNR improvement. The processing gain of spectral analysis depends on the window function, both its noise bandwidth (B) and its potential scalloping loss. These effects partially offset, because windows with the least scalloping naturally have the most leakage.

The figure at right depicts the effects of three different window functions on the same data set, comprising two equal strength sinusoids in additive noise. The frequencies of the sinusoids are chosen such that one encounters no scalloping and the other encounters maximum scalloping. Both sinusoids suffer less SNR loss under the Hann window than under the Blackman–Harris window. In general (as mentioned earlier), this is a deterrent to using high-dynamic-range windows in low-dynamic-range applications.

Filter design

Main article: 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.

A list of window functions

Terminology:

• N represents the width, in samples, of a discrete-time, symmetrical window function w(n). When N is an odd number, the non-flat windows have a singular maximum point. When N is even, they have a double maximum.
  • A common desire is for an asymmetrical window called DFT-even or periodic, which has a single maximum but an even number of samples (required by the FFT algorithm). Such a window would be generated by the Matlab function hann(512,'periodic'), for instance. Here, that window would be generated by N = 513 and discarding the 513^th element of the w(n) sequence.
• n is an integer, with values 0 ≤ n ≤ N − 1. Thus, these are lagged versions of unlagged functions denoted w₀(n) whose maximum occurs at n = 0:

$w(n) = w_0\left(n-\frac{N-1}{2}\right)$

• Each figure label includes the corresponding noise equivalent bandwidth metric (B), in units of DFT bins.

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 triangular window (k = 2) and the Parzen window (k = 4). Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows. A kth 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.

Rectangular window

Rectangular window; B = 1.0000.

The rectangular window (sometimes known as the boxcar or Dirichlet 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:

$w(n) = 1.$

Other windows are designed to moderate these sudden changes because discontinuities have undesirable effects on the discrete-time Fourier transform (DTFT) and/or the algorithms that produce samples of the DTFT.

The rectangular window is the 1st order B-spline window as well as the 0th power cosine window.

Triangular window

Triangular window or equivalently the Bartlett window; B = 1.3333. The two windows converge at large N as is the case here.

The triangular window is defined by:

$w(n)=1 - \left|\frac{n-\frac{N-1}{2}}{\frac{N+1}{2}}\right|$.

The end samples are positive (equal to 2/(N + 1)). This window can be seen as the convolution of two half-sized rectangular windows (for N even), giving it a main lobe width of twice the width of a regular rectangular window. The nearest lobe is −26 dB down from the main lobe.

The triangular window is the 2nd order B-spline window.

The Bartlett window is a slightly narrower variant of the triangular window, with zero weight at both ends:

$w(n)=1 - \left|\frac{n-\frac{N-1}{2}}{\frac{N-1}{2}}\right|$

Parzen window

Parzen window; B = 1.92.

Not to be confused with Kernel density estimation.

The Parzen window, also known as the de la Vallé Poussin window, is the 4th order B-spline window.

Other polynomial windows

Welch window

Welch window; B = 1.20.

The Welch window consists of a single parabolic section:

$w(n)=1 - \left(\frac{n-\frac{N-1}{2}}{\frac{N+1}{2}}\right)^2$.

The defining quadratic polynomial reaches a value of zero at the samples just outside the span of the window.

Generalized Hamming windows

Generalized Hamming windows are of the form:

$w(n) = \alpha - \beta\; \cos\left( \frac{2 \pi n}{N - 1} \right)\,$.

They have only three non-zero DFT coefficients and share the benefits of a sparse frequency domain representation with higher-order generalized cosine windows.

Hann (Hanning) window

Hann window; B = 1.5000.

Main article: Hann function

The Hann window named after Julius von Hann and also known as the Hanning (for being similar in name and form to the Hamming window), von Hann and the raised cosine window is defined by:

$w(n) = 0.5\; \left(1 - \cos \left ( \frac{2 \pi n}{N-1} \right) \right)$

• unlagged version:

$w_0(n) = 0.5\; \left(1 + \cos \left ( \frac{2 \pi n}{N-1} \right) \right)$

The ends of the cosine just touch zero, so the side-lobes roll off at about 18 dB per octave.

Hamming window

Hamming window, α = 0.53836 and β = 0.46164; B = 1.37. The original Hamming window would have α = 0.54 and β = 0.46; B = 1.3628.

The window with these particular coefficients was proposed by Richard W. Hamming. The window is optimized to minimize the maximum (nearest) side lobe, giving it a height of about one-fifth that of the Hann window.

$w(n) = \alpha - \beta\; \cos\left( \frac{2 \pi n}{N - 1} \right),$

with

$\alpha = 0.54,\; \beta = 1 - \alpha = 0.46,$

instead of both constants being equal to 1/2 in the Hann window. The constants are approximations of values α = 25/46 and β = 21/46, which cancel the first sidelobe of the Hann window by placing a zero at frequency 5π/(N − 1). Approximation of the constants 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 α = 0.53836 and β = 0.46164.

• Unlagged version:

$\begin{align} w_0(n)\ &\stackrel{\mathrm{def}}{=}\ w(n+\begin{matrix} \frac{N-1}{2}\end{matrix})\\ &= 0.54 + 0.46\; \cos \left ( \frac{2\pi n}{N-1} \right) \end{align}$

Higher-order generalized cosine windows

Windows of the form:

$w(n) = \sum_{k = 0}^{K} a_k\; \cos\left( \frac{2 \pi k n}{N} \right)$

have only 2K + 1 non-zero DFT coefficients, which makes them good choices for applications that require windowing by convolution in the frequency-domain. In those applications, the DFT of the unwindowed data vector is needed for a different purpose than spectral analysis. (see Overlap-save method). Generalized cosine windows with just two terms (K = 1) belong in the subfamily generalized Hamming windows.

Blackman windows

Blackman window; α = 0.16; B = 1.73.

Blackman windows are defined as:

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right) + a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)$

$a_0=\frac{1-\alpha}{2};\quad a_1=\frac{1}{2};\quad a_2=\frac{\alpha}{2}\,$

By common convention, the unqualified term Blackman window refers to α = 0.16, as this most closely approximates the "exact Blackman", with a₀ = 7938/18608 ≈ 0.42659, a₁ = 9240/18608 ≈ 0.49656, and a₂ = 1430/18608 ≈ 0.076849. These exact values place zeros at the third and fourth sidelobes.

Nuttall window, continuous first derivative

Nuttall window, continuous first derivative; B = 2.0212.

Considering n as a real number, the function and its first derivative are continuous everywhere.

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$

$a_0=0.355768;\quad a_1=0.487396;\quad a_2=0.144232;\quad a_3=0.012604\,$

Blackman–Nuttall window

Blackman–Nuttall window; B = 1.9761.

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$

$a_0=0.3635819; \quad a_1=0.4891775; \quad a_2=0.1365995; \quad a_3=0.0106411\,$

Blackman–Harris window

Blackman–Harris window; B = 2.0044.

A generalization of the Hamming family, produced by adding more shifted sinc functions, meant to minimize side-lobe levels

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)$

$a_0=0.35875;\quad a_1=0.48829;\quad a_2=0.14128;\quad a_3=0.01168\,$

Flat top window

SRS flat top window; B = 3.7702.

A flat top window is a partially negative-valued window that has a flat top in the frequency domain. Such windows have been made available in spectrum analyzers for the measurement of amplitudes of sinusoidal frequency components. They have a low amplitude measurement error suitable for this purpose, achieved by the spreading of the energy of a sine wave over multiple bins in the spectrum. This ensures that the unattenuated amplitude of the sinusoid can be found on at least one of the neighboring bins. The drawback of the broad bandwidth is poor frequency resolution. To compensate, a longer window length may be chosen.

Flat top windows can be designed using low-pass filter design methods, or they may be of the usual sum-of-cosine-terms variety. An example of the latter is the flat top window available in the Stanford Research Systems (SRS) SR785 spectrum analyzer:

$w(n)=a_0 - a_1 \cos \left ( \frac{2 \pi n}{N-1} \right)+ a_2 \cos \left ( \frac{4 \pi n}{N-1} \right)- a_3 \cos \left ( \frac{6 \pi n}{N-1} \right)+a_4 \cos \left ( \frac{8 \pi n}{N-1} \right)$

$a_0=1;\quad a_1=1.93;\quad a_2=1.29;\quad a_3=0.388;\quad a_4=0.028\,$

Rife–Vincent window

Rife and Vincent define three classes of windows constructed as sums of cosines; the classes are generalizations of the Hanning window. Their order-P windows are of the form (normalized to have unity average as opposed to unity max as the windows above are):

$w(n) = 1 + \sum_{l = 1}^P a_l \cos \left ( \frac{l 2 \pi n}{N-1} \right)$.

For order 1, this formula can match the Hanning window for a₁ = −1; this is the Rife–Vincent class-I window, defined by minimizing the high-order sidelobe amplitude. The class-I order-2 Rife–Vincent window has a₁ = −4/3 and a₂ = 1/3. Coefficients for orders up to 4 are tabulated. For orders greater than 1, the Rife–Vincent window coefficients can be optimized for class II, meaning minimized main-lobe width for a given maximum side-lobe, or for class III, a compromise for which order 2 resembles Blackmann's window. Given the wide variety of Rife–Vincent windows, plots are not given here.

Power-of-cosine windows

Window functions in the power-of-cosine family are of form:

$w(n) = \cos^\alpha\left(\frac{\pi n}{N-1} - \frac{\pi}{2}\right)$

The rectangular window (α = 0) and the Hann window (α = 2) are also members of this family.

Cosine window

Cosine window; B = 1.23.

$w(n) = \cos\left(\frac{\pi n}{N-1} - \frac{\pi}{2}\right) = \sin\left(\frac{\pi n}{N-1}\right)$

The cosine window is also known as the sine window. Cosine window describes the shape of $w_0(n)\,$

A cosine window convolved by itself is known as the Bohman window.

Adjustable windows

Gaussian window

Gaussian window, σ = 0.4; B = 1.45.

The Fourier transform of a Gaussian is also a Gaussian (it is an eigenfunction of the Fourier Transform). Since the 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, this can be used for exact quadratic interpolation in frequency estimation.

$w(n)=e^{-\frac{1}{2} \left ( \frac{n-(N-1)/2}{\sigma (N-1)/2} \right)^{2}}$

$\sigma \le \;0.5\,$

Tukey window

Tukey window, α = 0.5; B = 1.22.

The Tukey window, also known as the tapered cosine window, can be regarded as a cosine lobe of width αN/2 that is convolved with a rectangular window of width (1 − α/2)N.

$w(n) = \left\{ \begin{matrix} \frac{1}{2} \left[1+\cos \left(\pi \left( \frac{2 n}{\alpha (N-1)}-1 \right) \right) \right] & 0 \leqslant n \leqslant \frac{\alpha (N-1)}{2} \\ 1 & \frac{\alpha (N-1)}{2}\leqslant n \leqslant (N-1) (1 - \frac{\alpha}{2}) \\ \frac{1}{2} \left[1+\cos \left(\pi \left( \frac{2 n}{\alpha (N-1)}- \frac{2}{\alpha} + 1 \right) \right) \right] & (N-1) (1 - \frac{\alpha}{2}) \leqslant n \leqslant (N-1) \\ \end{matrix} \right.$

At α = 0 it becomes rectangular, and at α = 1 it becomes a Hann window.

Planck-taper window

Planck-taper window, ε = 0.1; B = 1.10.

The form of the Planck-taper window is inspired by the Planck distribution. It was first suggested in the context of gravitational-wave astronomy. It is defined as a piecewise function:

$w(n) = \left\{ \begin{matrix} \frac{1}{\exp(Z_+)+1} & 0 \leqslant n < \epsilon(N - 1) \\ 1 & \epsilon(N - 1) < n < (1 - \epsilon)(N - 1) \\ \frac{1}{\exp(Z_-)+1} & (1 - \epsilon)(N - 1) < n \leqslant (N - 1) \\ 0 & \mbox{otherwise} \\ \end{matrix} \right.$

where

$Z_\pm(n; \epsilon) = 2\epsilon\left[\frac{1}{1 \pm 2 n /(N - 1)} + \frac{1}{1 - 2\epsilon \pm 2 n / (N - 1)}\right].$

It transitions smoothly between 0 and 1, the amount of tapering is controlled by the parameter ε, with smaller values giving sharper transitions.

DPSS or Slepian window

DPSS window, α = 2; B = 1.47.

DPSS window, α = 3; B = 1.77.

The DPSS (discrete prolate spheroidal sequence) or Slepian window is used to maximize the energy concentration in the main lobe.

The main lobe ends at a bin given by the parameter α.

Kaiser window

Kaiser window, α = 2; B = 1.4963.

Kaiser window, α = 3; B = 1.7952.

Main article: Kaiser window

The Kaiser, or Kaiser-Bessel, window is a simple approximation of the DPSS window using Bessel functions, discovered by Jim Kaiser.

$w(n)=\frac{I_0\left(\pi\alpha \sqrt{1-(\frac{2 n}{N-1}-1)^2}\right)}{I_0(\pi\alpha)}$

where I₀ is the zero-th order modified Bessel function of the first kind, and usually α = 3.

The null that ends the main lobe is located at bin $\sqrt{1 + \alpha^2}$.

• unlagged version:

$w_0(n) = \frac{I_0\left(\pi\alpha \sqrt{1-(\frac{2 n}{N-1})^2}\right)}{I_0(\pi\alpha)}$

Dolph–Chebyshev window

Dolph–Chebyshev window, α = 5; B = 1.94.

Minimizes the Chebyshev norm of the side-lobes for a given main lobe width.

The unlagged Dolph–Chebyshev window function w₀(n) is defined in terms of its real-valued discrete Fourier transform, W₀(k):

$\begin{align} W_0(k) &= \frac{\cos\{N \cos^{-1}[\beta \cos(\frac{\pi k}{N})]\}}{\cosh[N \cosh^{-1}(\beta)]}\\ \beta &= \cosh[\frac{1}{N} \cosh^{-1}(10^\alpha)] \end{align}$

where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.

The window function can be calculated from W₀(k) through inverse discrete Fourier transform:

$w_0(n) = \frac{1}{N} \sum_{k=0}^{N-1} W_0(k) \cdot e^{i 2 \pi k n / N}$

The lagged version of the window, with 0 ≤ n ≤ N−1, can be obtained via:

$w(n) = w_0\left(n-\frac{N-1}{2}\right)$.

Exponential or Poisson window

Exponential window, τ = N/2, B = 1.08.

Exponential window, τ = (N/2)/(60/8.69), B = 3.46.

The Poisson window, or more generically the exponential window increases exponentially towards the center of the window and decreases exponentially in the second half. Since the exponential function never reaches zero, the values of the window at its limits are non-zero (it can be seen as the multiplication of an exponential function by a rectangular window ). It is defined by

$w(n)=e^{-\left|n-\frac{N-1}{2}\right|\frac{1}{\tau}},$

where τ is the time constant of the function. The exponential function decays as e ≃ 2.71828 or approximately 8.69 dB per time constant. This means that for a targeted decay of D dB over half of the window length, the time constant τ is given by

$\tau = \frac{N}{2}\frac{8.69}{D}.$

Hybrid windows

Window functions have also been constructed as multiplicative or additive combinations of other windows.

Bartlett–Hann window

Bartlett–Hann window; B = 1.46.

$w(n)=a_0 - a_1 \left |\frac{n}{N-1}-\frac{1}{2} \right| - a_2 \cos \left (\frac{2 \pi n}{N-1}\right )$

$a_0=0.62;\quad a_1=0.48;\quad a_2=0.38\,$

Planck–Bessel window

Planck–Bessel window, ε = 0.1, α = 4.45; B = 2.16.

A Planck-taper window multiplied by a Kaiser window which is defined in terms of a modified Bessel function. This hybrid window function was introduced to decrease the peak side-lobe level of the Planck-taper window while still exploiting its good asymptotic decay. It has two tunable parameters, ε from the Planck-taper and α from the Kaiser window, so it can be adjusted to fit the requirements of a given signal.

Hann–Poisson window

Hann–Poisson window, α = 2; B = 2.02

A Hann window multiplied by a Poisson window, which has no side-lobes, in the sense that its Fourier transform drops off forever away from the main lobe. It can thus be used in hill climbing algorithms like Newton's method. The Hann–Poisson window is defined by:

$w(n)=\frac{1}{2}\left(1-\cos\left(\frac{2 \pi n}{N - 1}\right)\right)e^\frac{-\alpha\left|N - 1 - 2n\right|}{N - 1}\,$

where α is a parameter that controls the slope of the exponential.

Other windows

Lanczos window

Sinc or Lanczos window; B = 1.30.

$w(n) = \mathrm{sinc}\left(\frac{2n}{N-1}-1\right)$

• used in Lanczos resampling
• for the Lanczos window, sinc(x) is defined as sin(πx)/(πx)
• also known as a sinc window, because:

$w_0(n) = \mathrm{sinc}\left(\frac{2n}{N-1}\right)\,$ is the main lobe of a normalized sinc function

Comparison of windows

Window functions in the frequency domain ("spectral leakage")

When selecting an appropriate window function for an application, this comparison graph may be useful. The frequency axis has units of FFT "bins" when the window of length N is applied to data and a transform of length N is computed. For instance, the value at frequency ½ "bin" (third tick mark) is the response that would be measured in bins k and k+1 to a sinusoidal signal at frequency k+½. It is relative to the maximum possible response, which occurs when the signal frequency is an integer number of bins. The value at frequency ½ is referred to as the maximum scalloping loss of the window, which is one metric used to compare windows. The rectangular window is noticeably worse than the others in terms of that metric.

Other metrics that can be seen are the width of the main lobe and the peak level of the sidelobes, which respectively determine the ability to resolve comparable strength signals and disparate strength signals. The rectangular window (for instance) is the best choice for the former and the worst choice for the latter. What cannot be seen from the graphs is that the rectangular window has the best noise bandwidth, which makes it a good candidate for detecting low-level sinusoids in an otherwise white noise environment. Interpolation techniques, such as zero-padding and frequency-shifting, are available to mitigate its potential scalloping loss.

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 utilized in, e.g., image processing. They can be constructed from one-dimensional windows in either of two forms.

The separable form, $\scriptstyle W(m,n)=w(m)w(n)$ is trivial to compute. The radial form, $\scriptstyle W(m,n)=w(r)$, which involves the radius $\scriptstyle r=\sqrt{(m-M/2)^2+(n-N/2)^2}$, 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 appertures, which can be visualized in terms of the product of two sinc functions vs. an Airy function, respectively.

See also

• Spectral leakage
• Multitaper
• Apodization
• Welch method
• Short-time Fourier transform
• Window design method
• Kolmogorov–Zurbenko filter

Further reading

• Nuttall, Albert H. (February 1981). "Some Windows with Very Good Sidelobe Behavior". IEEE Transactions on Acoustics, Speech, and Signal Processing 29 (1): 84–91. doi:10.1109/TASSP.1981.1163506.  Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.

• Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John A. (1999). Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall. pp. 468–471. ISBN 0-13-754920-2. 

• Bergen, S.W.A.; A. Antoniou (2004). "Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics". EURASIP Journal on Applied Signal Processing 2004 (13): 2053–2065. doi:10.1155/S1110865704403114. 

• Bergen, S.W.A.; A. Antoniou (2005). "Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function". EURASIP Journal on Applied Signal Processing 2005 (12): 1910–1922. doi:10.1155/ASP.2005.1910. 

• US patent 7065150, Park, Young-Seo, "System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation", published 2003, issued 2006 

• Albrecht, Hans-Helge (2012). Tailored minimum sidelobe and minimum sidelobe cosine-sum windows. A catalog. doi:10.7795/110.20121022aa. 

External links

• LabView Help, Characteristics of Smoothing Filters, http://zone.ni.com/reference/en-XX/help/371361B-01/lvanlsconcepts/char_smoothing_windows/

• Evaluation of Various Window Function using Multi-Instrument, http://www.multi-instrument.com/doc/D1003/Evaluation_of_Various_Window_Functions_using_Multi-Instrument_D1003.pdf