For the term used in SQL statements, see. In, a window function (also known as an apodization function or tapering function ) is a that is zero-valued outside of some chosen. 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'. In typical applications, the window functions used are non-negative, smooth, 'bell-shaped' curves. Rectangle, triangle, and other functions can also be used.

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• Dolph Chebyshev Array

Matlab Program For Dolph Chebyshev Array In Java. Β = cos 1 / N cosh − 1 ( 10 α ) α determines the level of the sidelobe attenuation. The level of the sidelobe attenuation is equal to − 20 α. For example, 100 dB of attenuation results from setting α = 5 The discrete-time Dolph-Chebyshev window is obtained by taking the inverse DFT.

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, and, more specifically, that the function goes sufficiently rapidly toward zero. Contents.

Applications Applications of window functions include /modification/, the design of filters, as well as and design. Spectral analysis The of the function cos ω t 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. Figure 1: Zoomed view of spectral leakage Windowing Windowing of a simple waveform like cos ω t causes its Fourier transform to develop non-zero values (commonly called ) 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. 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 and. 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. 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 (DFT).

But the DFT provides only a sparse 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 coincides 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 ½ bin), the 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.

Figure 2: 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. It can be thought of as redistributing the DTFT into a rectangular shape with height equal to the spectral maximum and width B.

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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, averaged over time, typically reveals a flat, 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, 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 (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 – 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: Window functions are sometimes used in the field of to restrict the set of data being analyzed to a range near a given point, with a that diminishes the effect of points farther away from the portion of the curve being fit. In the field of Bayesian analysis and, this is often referred to as the. Rectangular window applications Analysis of transients When analyzing a transient signal in, 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. Harmonic analysis 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 1, 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.

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 environment. Interpolation techniques, such as and frequency-shifting, are available to mitigate its potential scalloping loss. Albrecht, Hans-Helge (2012).

Dolph Chebyshev Array

Tailored minimum sidelobe and minimum sidelobe cosine-sum windows. A.; Antoniou, A. 'Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics'. EURASIP Journal on Applied Signal Processing. 2004 (13): 2053–2065. A.; Antoniou, A.

'Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function'. EURASIP Journal on Applied Signal Processing.

2005 (12): 1910–1922. Nuttall, Albert H. (February 1981). 'Some Windows with Very Good Sidelobe Behavior'. IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91.

Extends Harris' paper, covering all the window functions known at the time, along with key metric comparisons.;; Buck, John A. Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall., Park, Young-Seo, 'System and method for generating a root raised cosine orthogonal frequency division multiplexing (RRC OFDM) modulation', published 2003, issued 2006 External links. LabView Help, Characteristics of Smoothing Filters,.

Evaluation of Various Window Function using Multi-Instrument,. Creation and properties of Cosine-sum Window functions.