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. Mathematically, 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". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.
The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution. But that method also changes the frequency content of the signal by an effect called spectral leakage. Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.
In typical applications, the window functions used are non-negative, smooth, "bell-shaped" curves. Rectangle, triangle, and other functions can also be used. A rectangular window does not modify the data segment at all. It's only for modelling purposes that we say it multiplies by 1 inside the window and by 0 outside. 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.
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 a similar transform) 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.
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 stronger 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 and sensitivity, which is the ability to reveal relatively weak sinusoids in the presence of additive random noise. That is because the noise produces a stronger response with high-dynamic-range windows than with high-resolution windows. Therefore, high-dynamic-range windows are 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 trade-off between resolving comparable strength components with similar frequencies and resolving disparate strength components with dissimilar frequencies. That trade-off occurs when the window function is chosen.
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 sparse sampling of the actual discrete-time Fourier transform (DTFT) spectrum. Figure 2, row 3 shows a DTFT for a rectangularly-windowed sinusoid. The actual frequency of the sinusoid is indicated as "13" 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 coincides with a DFT sample, and the maximum value of the spectrum is accurately measured by that sample. In row 4, it misses the maximum value by ½ bin, and the resultant measurement error is referred to as scalloping loss (inspired by the shape of the peak). For a known frequency, such as a musical note or a sinusoidal test signal, matching the frequency to a DFT bin can be prearranged by choices of a sampling rate and a window length that results in an integer number of cycles within the window.
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. It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B. The more the 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.
Figure 3 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.
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.
Statistics and curve fitting
Window functions are sometimes used in the field of statistical analysis to restrict the set of data being analyzed to a range near a given point, with a weighting factor that diminishes the effect of points farther away from the portion of the curve being fit. In the field of Bayesian analysis and curve fitting, this is often referred to as the kernel.
Rectangular window applications
Analysis of transients
When analyzing a transient signal in modal analysis, such as an impulse, a shock response, a sine burst, a chirp burst, or noise burst, where the energy vs time distribution is extremely uneven, the rectangular window may be most appropriate. For instance, when most of the energy is located at the beginning of the recording, a non-rectangular window attenuates most of the energy, degrading the signal-to-noise ratio.
One might wish to measure the harmonic content of a musical note from a particular instrument or the harmonic distortion of an amplifier at a given frequency. Referring again to Figure 2, we can observe that there is no leakage at a discrete set of harmonically-related frequencies sampled by the DFT. (The spectral nulls are actually zero-crossings, which cannot be shown on a logarithmic scale such as this.) This property is unique to the rectangular window, and it must be appropriately configured for the signal frequency, as described above.
The formulas provided in this article produce discrete sequences, as if a continuous window function has been "sampled". (see an example at Kaiser window) Window sequences can be either symmetric or 1-sample short of symmetric (called asymmetric or periodic). For instance, a symmetric sequence, with its maximum at a single center-point, is generated by the MATLAB function
hann(9,'symmetric'). Deleting the last sample produces a sequence identical to
hann(8,'periodic'). An even-length symmetric sequence has two equal center-points, but most window functions used in practice have a single peak value, whether they are symmetric or asymmetric.
Some functions have one or two zero-valued end-points, which are unnecessary in most applications. Deleting a zero-valued end-point has no effect on its DTFT (spectral leakage). But the function designed for N+1 samples, in anticipation of deleting an end point, typically has a slightly narrower main lobe, slightly higher sidelobes, and a slightly lower noise bandwidth. Similarly, deleting both zeros from a function designed for N+2 samples further enhances those effects.
The terminology DFT-even describes a subclass of periodic (or asymmetric) windows, characterized by only even-length sequences. Their potential advantages (application-dependent) for spectral analysis (DFT) are three-fold:
- Efficiency: FFT algorithms are most efficient when the sequence length is an integer power-of-two.
- Real-valued DFT coefficients: The phase components of the DFT are due to the windowed data, not the windowing function. Although there are sign reversals (which can appear as a 180° phase-shift), they are predictable, as explained below.
- Zero-valued DFT coefficients: For several popular window functions, including the #Rectangular window and #Cosine-sum windows, most of the DFT coefficients are zero-valued. We see the rectangular window effect in the third row of Figure 2. A cosine-sum example is: DFT-even Hann window, which shows that the N-point DFT of the sequence generated by hann(N,'periodic') has only 3 non-zero values. All the other samples coincide with zero-crossings of the DTFT. Besides the application already described at #Harmonic analysis, this property is useful for real-time applications that require both windowed and non-windowed (rectangularly windowed) transforms, because the windowed transforms can be efficiently derived from the non-windowed transforms by convolution. The real-valued coefficients are also an efficiency multiplier.
When a symmetric sequence is associated with the indices [−M ≤ n ≤ M], known as a finite Fourier transform data window, its DTFT, a continuous function of frequency is real-valued. When the sequence is shifted into a DFT data window, [0 ≤ n ≤ 2M], the DTFT is multiplied by a complex-valued phase function: . But when sampled at frequencies for integer values of the samples are all real-valued. The exact values of samples at intervals of 1/2M are easily computed by a 2M-length DFT, where the last windowed data value is added to the first. (See Sampling the DTFT, Case L = N + 1.) And because of window symmetry, that is equivalent to adding the two unwindowed data values before applying the window function. Effectively, only the first 2M window coefficients (a DFT-even window) are used. As a practical matter, adding the last and first samples is usually more trouble than it's worth, because of the small values of the first window coefficient. So that step is commonly omitted, even though deleting an end-point can have undesirable effects on a window's sidelobe structure. An example can be seen at File:Comparison of symmetric and periodic Gaussian windows.svg, where the 9-sample symmetric sequence (green DTFT) has better spectral leakage metrics than the 8-sample truncated sequence (blue). But for typically large window lengths, the effect will not be as pronounced.
A list of window functions
- is a zero-phase (symmetrical about x=0) function, continuous for where N is a positive integer (even or odd).
- The sequence is symmetric, of length
- is asymmetric, of length
- The parameter B displayed on each spectral plot is the function's noise equivalent bandwidth metric, in units of DFT bins.
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:
Other windows are designed to moderate these sudden changes, which reduces scalloping loss and improves dynamic range, as described above (§ Spectral analysis).
The rectangular window is the 1st order B-spline window as well as the 0th power Power-of-sine window.
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.
Triangular windows are given by:
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.
Other polynomial windows
The Welch window consists of a single parabolic section:
The defining quadratic polynomial reaches a value of zero at the samples just outside the span of the window.
The corresponding function is a cosine without the π/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.
This family is also known as generalized cosine windows.
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, and sometimes 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, 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 5π/(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.
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 a0 = 0.53836 and a1 = 0.46164.
Blackman windows are defined as:
By common convention, the unqualified term Blackman window refers to Blackman's "not very serious proposal" of α = 0.16 (a0 = 0.42, a1 = 0.5, a2 = 0.08), which 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, 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 Nuttall window, and its first derivative are continuous everywhere, like the Hann function. That is, the function goes to 0 at x= ±N/2, unlike the Blackman–Nuttall, Blackman–Harris, and Hamming windows. The Blackman window (α = 0.16) is also continuous with continuous derivative at the edge, but the "exact Blackman window" is not.
Flat top window
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 #Noise bandwidth.
Flat top windows can be designed using low-pass filter design methods, or they may be of the usual cosine-sum variety:
The Matlab variant has these coefficients:
Rife–Vincent windows are customarily scaled for unity average value, instead of unity peak value. The coefficient values below, applied to Eq.1, reflect that custom.
Class I, Order 1 (K=1): Functionally equivalent to the Hann window.
Class I, Order 2 (K=2):
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 #Blackman window.
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.
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 σω for a given temporal width (N+1)σt. 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 #Sine window and the #Gaussian window in the limiting cases of large and small σt, respectively.
Approximate confined Gaussian window
Defining L ≜ N+1, a #Confined Gaussian window of temporal width L × σt is well approximated by:
where is a Gaussian function:
The standard deviation of the approximate window is asymptotically equal (i.e. large values of N) to L × σt for σt < 0.14.
Generalized normal window
A more generalized version of the Gaussian window is the generalized normal window. Retaining the notation from the Gaussian window 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 Tukey window discussed later, 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).
Defining L ≜ N+1, the Tukey window, also known as the cosine-tapered window, can be regarded as a cosine lobe of width Lα/2 that is convolved with a rectangular window of width L(1−α/2).
At α = 0 it becomes rectangular, and at α = 1 it becomes a Hann window.
The so-called "Planck-taper" window is a bump function that has been widely used in the theory of partitions of unity in manifolds. It is smooth (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, inspired by the Planck distribution. It is defined as a piecewise 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 Kaiser windows below are created by a simple approximation to the DPSS windows:
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. Sometimes the Kaiser window is parametrized by β, where β = πα.
Minimizes the Chebyshev norm of the side-lobes for a given main lobe width.
Tn(x) is the n-th Chebyshev polynomial of the first kind evaluated in x, which can be computed using
The window function can be calculated from W0(k) by an inverse discrete Fourier transform (DFT):
The lagged version of the window can be obtained by:
which for even values of N must be computed as follows:
which is an inverse DFT of
- Due to the equiripple condition, the time-domain window has discontinuities at the edges. An approximation that avoids them, by allowing the equiripples to drop off at the edges, is a Taylor window.
- An alternative to the inverse DFT definition is also available..
Like other adjustable windows, the Ultraspherical window has parameters that can be used to control its Fourier transform main-lobe width and relative side-lobe amplitude. Uncommon to other windows, it has an additional parameter which can be used to set the rate at which side-lobes decrease (or increase) in amplitude.
where is the Ultraspherical polynomial of degree N, and and control the side-lobe patterns.
Certain specific values of yield other well-known windows: and give the Dolph–Chebyshev and Saramäki windows respectively. See here for illustration of Ultraspherical windows with varied parametrization.
Exponential or Poisson window
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
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
Window functions have also been constructed as multiplicative or additive combinations of other windows.
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.
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:
where α is a parameter that controls the slope of the exponential.
Comparison of windows
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.
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 are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform. They can be constructed from one-dimensional windows in either of two forms. The separable form, is trivial to compute. The radial form, , which involves the radius , 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.
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- Mathematically, the noise equivalent bandwidth of transfer function H is the bandwidth of an ideal rectangular filter with the same peak gain as H that would pass the same power with white noise input. In the units of frequency f (e.g. hertz), it is given by:
- Periodic is an oblique reference to the fact that symmetry around the n=0 axis is restored when the sequence is viewed as one full cycle of a periodic sequence.
- The sparse sampling of a DTFT 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. Thus, when choosing a window function, it is usually important to sample the DTFT more densely (as we do in the next section) and choose a window that suppresses the sidelobes to an acceptable level.
- The N-point DFT of an N-sample DFT-even Hann or Hamming window, for example, has only 3 DTFT samples that do not coincide with zero-crossings. An illustration, for N=16, can be viewed at DFT-even Hann window.
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Note that, on a dB scale, Gaussians are quadratic. This means that parabolic interpolation of a sampled Gaussian transform is exact. ... quadratic interpolation of spectral peaks may be more accurate on a log-magnitude scale (e.g., dB) than on a linear magnitude scale
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This family of window functions was "discovered" by Kaiser in 1962 following a discussion with B. F. Logan of the Bell Telephone Laboratories. ... Another valuable property of this family ... is that they also approximate closely the prolate spheroidal wave functions of order zero
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Sometimes the Kaiser window is parametrized by α, where β=πα.
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