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.
Choice of window functionEdit
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 our ability to distinguish them spectrally. Possible types of interference are often broken down into two opposing classes as follows: If the component 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 too similar, leakage can render them unresolvable even when the sinusoids are of equal strength. Windows that are effective against the first type of interference, namely where components have dissimilar frequencies and amplitudes, are called high dynamic range. Conversely, windows that can distinguish components with similar frequencies and amplitudes are called high resolution.
The rectangular window is an example of a window that is high resolution but low dynamic range, meaning it is good for distinguishing components of similar amplitude even when the frequencies are also close, but poor at distinguishing components of different amplitude even when the frequencies are far away. High-resolution, low-dynamic-range windows such as the rectangular window also have the property of high 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.
At the other extreme of the range of window types are windows with high dynamic range but low resolution and sensitivity. 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 summary, spectral analysis involves a trade-off between resolving comparable strength components with similar frequencies (high resolution / sensitivity) and resolving disparate strength components with dissimilar frequencies (high dynamic range). That trade-off occurs when the window function is chosen.:p.90
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.[A] 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 lossesEdit
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 fittingEdit
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 applicationsEdit
Analysis of transientsEdit
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).[B] 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 or N+2 samples, in anticipation of deleting one or both end points, typically has a slightly narrower main lobe, slightly higher sidelobes, and a slightly smaller noise-bandwidth.
The symmetry of window functions offers the possibility of a DTFT that is entirely real-valued. That requires an odd-length (N+1) symmetric sequence, centered at the origin of the x (or time) axis. When the same sequence is shifted to begin at the origin, placing it in a DFT data window, the DTFT becomes complex-valued, except at frequencies spaced at regular intervals of 1/N.[a] Thus, when sampled by an N-length DFT, the samples (called DFT coefficients) are still real-valued. The customary method is by truncating the sequence, which creates an asymmetric window and changes the DTFT (spectral leakage), usually by a negligible amount (when N is large).[C] The even-length, asymmetric sequence has been called a DFT-even window function.:p.52
- Another method, which does not affect the DTFT being sampled, is to compute the DFT on one cycle of a periodic summation (period = N) of the symmetric sequence. I.e. the last of N+1 samples is combined with the first one by addition and deleted, leaving an N-length sequence.[D][E]
- An odd-length, asymmetric window also has real-valued DFT coefficients. But even-length windows are preferable when using the FFT algorithm to implement the DFT calculation.
- When these truncated windows are applied to actual data, the sequence will generally lack symmetry, and the DFT will not be real-valued.
Despite these caveats, some authors automatically assume DFT-even windows.[b] So it is worth noting that there is no performance advantage when applied to time domain data, which is the customary application. The advantage of real-valued DFT coefficients is realized in certain esoteric applications[F] where windowing is achieved by means of convolution between the DFT coefficients and an unwindowed DFT of the data.:p.62:p.85 In those applications, asymmetric windows from the Cosine-sum family are preferred, because most of their DFT coefficients are zero-valued, making the convolution very efficient.[G]:p.85 Odd-length, asymmetric windows also have that attribute.[H]
A list of window functionsEdit
- is a zero-phase function (symmetrical about x=0), continuous for where N is a positive integer (even or odd).
- The sequence is symmetric, of length
- is asymmetric, of length [I]
- 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 2) 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.[c] 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.
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.
Defining L ≜ N+1, the Parzen window, also known as the de la Vallée Poussin window, is the 4th order B-spline window given by:
Other polynomial windowsEdit
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.
These window functions have the form:
This family is also known as generalized cosine windows.
In most cases, including the examples below, all coefficients ak ≥ 0. These windows have only 2K + 1 non-zero N-point DFT coefficients.
Hann and Hamming windowsEdit
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 derivativeEdit
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 windowEdit
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.
The Matlab variant has these coefficients:
Other variations are available, such as sidelobes that roll off at the cost of higher values near the main lobe.
Class I, Order 1 (K=1): Functionally equivalent to the Hann window.
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.
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 windowEdit
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 windowEdit
where is a Gaussian function:
Generalized normal windowEdit
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, 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 windowEdit
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:
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.
The zero-phase Dolph–Chebyshev window function is usually defined in terms of its real-valued discrete Fourier transform, :
Tn(x) is the n-th Chebyshev polynomial of the first kind evaluated in x, which can be computed using
is the unique positive real solution to , where the parameter α sets the Chebyshev norm of the sidelobes to −20α decibels.
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.
The window can be expressed in the time-domain as follows:
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 windowEdit
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.
- used in Lanczos resampling
- for the Lanczos window, is defined as
- also known as a sinc window, because:
- is the main lobe of a normalized sinc function
Comparison of windowsEdit
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.
- Truncation affects only the real components of the DTFT samples. The imaginary ones remain zero-valued. Harris attributes this effect to what he calls DFT-even symmetry. But it is actually a property of any length asymmetric window function.
- Figure Sampling the DTFT is an example of both methods, also showing how real-valued samples are obtained from the complex DTFT.
- An example of the effect of truncation on spectral leakage (in decibels) is figure Gaussian windows. The graph labeled DTFT periodic8 is the DTFT of the truncated window labeled periodic DFT-even (both blue). The green graph labeled DTFT symmetric9 corresponds to the same window with its symmetry restored. The DTFT samples, labeled DFT8 periodic summation, are an example of using periodic summation to sample it at the same frequencies as the blue graph.
- Sometimes both a windowed and an unwindowed (rectangularly windowed) DFT are needed.
- For example, see figure 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.
- For example, see figure DFT-odd Hann window.
- Some authors limit their attention to this important subset and to even values of N. But the window coefficient formulas are still the ones presented here.
- The Kaiser window is often parametrized by β, where β = πα.:p.474 The alternative use of just α facilitates comparisons to the DPSS windows.
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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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The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of α as the bin number of the edge of the critically sampled window main lobe.
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