Bruger:Peh-herlev/sandkasse-Vinduesfunktion

Type af matematiske funktioner som typisk anvendes inden for signalbehandling. Vinduesfunktioner bruges sammen med signaler i tidsdomæne (Som signalet ser ud på et Oscilloskop).

Vinduesfunktionen kan anvendes ved konstruktionen af digitale filtre og ved beregning af frekvensindhold af et signal fouriertransformation (DFT, FFT).

Ved at tilføje (multiplicere) en vinduesfunktion på et signal tilfører man vinduesfunktionens frekvens respons og bestemmer derved selektiviteten og side-”laper” (engelsk lobs). For at forstå konsekvensen af vinduesfunktionen er man nødt til at forstå sammenhænget mellem enhedsrespons og frekvensindhold (Laplacetransformation).

Vinduesfunktionseksempler

Terminology:

• $N\,$ Repræsenterer bredden, i samples, i et diskrete tids system. Når N er et ulige antal, vil et ikke fladt vindue have et enkelt toppunkt. Når N er lige, har det et dobbelt toppunkt.
• Hver figur tekst inkluderer tidsfunktionen, den resulterende frekvens respons og ækvivalente støjbåndbredden (B), i DFT bins enheder.

Rektangulærlvindue

Rectangular-vindue; B=1.00

Det rektangulærlvindue kaldes sometider også for Diracs deltafunktion. Det har værdien 1 for alle værdier fra n=0 til n=N. For værdier af n < 0 eller n > N er værdien 0.

$w(n)=1\,$

Trekant-vindue

Bartlett-vindue; B=1.33

Trekant-vindue med nul i enderne:

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

Triangular-vindue; B=1.33

Uden nul i enderne:

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

Hanning-vinduet (Hann-vindue)

Hanning vindue; frekvens respons

Hanning-vinduet (eller Hanning vinduesfunktion) er en matematisk funktion der bruges indenfor digital signalbehandling. Den er opkaldt efter Julius Ferdinand von Hann. Dets matematisk form er

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

Hamming-vinduet

Hamming-vindue; B=1,37.

Hamming-vinduet (eller Hamming vinduesfunktion) er en matematisk funktion der bruges indenfor digital signalbehandling. Den er opkaldt efter amerikaneren Richard Hamming. Dets matematisk form er

$w(t)=w(-t)=0.54+0.46\cos(t\pi /T)\qquad {\mbox{For}}\,0\leq t\leq T$

Hanning-vinduet er en funktion der har næsten sammen matematisk form, mens andre vinduesfunktioner er det rektangulære vindue, det triangulære vindue og Kaiser-vinduet. I forhold til det rektangulære og det triangulære vindue har Hamming-vinduet forholdsvis små "sidelapper" (engelsk: side lobes).

Tukey-vinduer

Tukey-vindue, α=0.5; B=1.22

$w(n)=\left\{{\begin{matrix}{\frac {1}{2}}\left[1+\cos \left(\pi \left({\frac {2n}{\alpha (N-1)}}-1\right)\right)\right]&{\mbox{when}}\,0\leqslant n\leqslant {\frac {\alpha (N-1)}{2}}\\[0.5em]1&{\mbox{when}}\,{\frac {\alpha (N-1)}{2}}\leqslant n\leqslant (N-1)(1-{\frac {\alpha }{2}})\\[0.5em]{\frac {1}{2}}\left[1+\cos \left(\pi \left({\frac {2n}{\alpha (N-1)}}-{\frac {2}{\alpha }}+1\right)\right)\right]&{\mbox{when}}\,(N-1)(1-{\frac {\alpha }{2}})\leqslant n\leqslant (N-1)\\\end{matrix}}\right.$

Tukey-vinduer, også kendt som tapered cosine window, kan betragtes som et Hanning-vindue med breden ${\tfrac {\alpha N}{2}}$ som er kombineret med et Rektangulærlvindue med breden $\left(1-{\tfrac {\alpha }{2}}\right)N.$  Ved α=0 bliver til det et Rektangulærlvindue, og ved α=1 bliver det et Hanning-vindue.

Cosine-vinduet

Cosine-vindue; B=1.23

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

• også kendt som sine window
• cosine-vindue $w_{0}(n)\,$

Lanczos-vinduet

Sinc eller Lanczos-vindue; B=1.30

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

• brugt i [Lanczos resampling]
• for Lanczos-vinduet, sinc(x) er defineret som sin(πx)/(πx)
• også kendt som sinc window, fordi :

$w_{0}(n)=\mathrm {sinc} \left({\frac {2n}{N-1}}\right)\,$ is the main lobe of a normalized [sinc function]

Gaussian-vinduer

Gauss-vindue, σ=0.4; B=1.45

The frequency response 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.

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

$\sigma \leq \;0.5\,$

Bartlett–Hann window

Bartlett-Hann window; B=1.46

$w(n)=a_{0}-a_{1}\left|{\frac {n}{N-1}}-{\frac {1}{2}}\right|-a_{2}\cos \left({\frac {2\pi n}{N-1}}\right)$

$a_{0}=0.62;\quad a_{1}=0.48;\quad a_{2}=0.38\,$

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.

Kaiser windows

Kaiser window, α =2; B=1.5

Kaiser window, α =3; B=1.8

Hovedartikel: Kaiser window.

A simple approximation of the DPSS window using Bessel functions, discovered by Jim Kaiser.

$w(n)={\frac {I_{0}{\Bigg (}\pi \alpha {\sqrt {1-({\begin{matrix}{\frac {2n}{N-1}}\end{matrix}}-1)^{2}}}{\Bigg )}}{I_{0}(\pi \alpha )}}$

where $I_{0}$ is the zero-th order modified Bessel function of the first kind, and usually $\alpha =3$.

• Note that:

$w_{0}(n)={\frac {I_{0}{\Bigg (}\pi \alpha {\sqrt {1-({\begin{matrix}{\frac {2n}{N-1}}\end{matrix}})^{2}}}{\Bigg )}}{I_{0}(\pi \alpha )}}$

Low-resolution (high-dynamic-range) windows

Nuttall window, continuous first derivative

Nuttall window, continuous first derivative; B=2.02

$w(n)=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)-a_{3}\cos \left({\frac {6\pi n}{N-1}}\right)$

$a_{0}=0.355768;\quad a_{1}=0.487396;\quad a_{2}=0.144232;\quad a_{3}=0.012604\,$

Blackman–Harris window

Blackman–Harris window; B=2.01

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

$w(n)=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)-a_{3}\cos \left({\frac {6\pi n}{N-1}}\right)$

$a_{0}=0.35875;\quad a_{1}=0.48829;\quad a_{2}=0.14128;\quad a_{3}=0.01168\,$

Blackman–Nuttall window

Blackman–Nuttall window; B=1.98

$w(n)=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)-a_{3}\cos \left({\frac {6\pi n}{N-1}}\right)$

$a_{0}=0.3635819;\quad a_{1}=0.4891775;\quad a_{2}=0.1365995;\quad a_{3}=0.0106411\,$

Flat top window

Flat top window; B=3.77

$w(n)=a_{0}-a_{1}\cos \left({\frac {2\pi n}{N-1}}\right)+a_{2}\cos \left({\frac {4\pi n}{N-1}}\right)-a_{3}\cos \left({\frac {6\pi n}{N-1}}\right)+a_{4}\cos \left({\frac {8\pi n}{N-1}}\right)$

$a_{0}=1;\quad a_{1}=1.93;\quad a_{2}=1.29;\quad a_{3}=0.388;\quad a_{4}=0.032\,$

Hann-Poisson window

A Hann window multiplied by a Poisson window, which has no side-lobes, in the sense that the frequency response drops off forever away from the main lobe. It can thus be used in hill climbing algorithms like Newton's method.

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 $\tau$ 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 $\tau$ is given by

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

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.

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.

Vindue funktion og fouriertransformation

En hoved anvendelse for vinduesfuntioner er sammen med fouriertransformation, uanset om det drejer sig om diskret fouriertransformation (FFT / DFT).

Illustration af Hanning-vindue ganget på signal før FFT funktion.

Tabel over vindues-funktioner

En given vindues-funktion påvirker det beregnet spektrum.

Oversigt over sammenhæng mellem vindues-funktion og selektivitet
Vidues-funktion		højeste "sidelapper" (lobes) [dB]	"sidelapper" fald [dB/okt]	Forstærkning [bin]	Støjbåndbrede [bin]	(-3dB) båndbrede [bin]	(-6dB) båndbrede [bin]
Retangulær		-13	-6	1.0	1.0	0.89	1.21
Trekant		-27	-12	0.5	1.33	1.28	1.78
Cos		-23	-12	0.64	1.23	1.20	1.65
Hanning (Cos^2)		-32	-18	0.5	1.50	1.44	2.00
Cos^3		-39	-24	0.42	1.73	1.66	2.32
Cos^4		-47	-30	0.38	1.94	1.86	2.59
Hamming		-43	-6	0.54	1.36	1.30	1.81
Riesz		-21	-12	0.67	1.20	1.16	1.59
Riemann		-26	-12	0.59	1.30	1.26	1.74
De La Valle-poussin		-53	-24	0.38	1.92	1.82	2.55
Tukey	a = 0.25 a = 0.50 a = 0.75	-14 -15 -19	-18 -18 -18	0.88 0.75 0.63	1.10 1.22 1.36	1.01 1.15 1.31	1.38 1.57 1.80
Bohman		-46	-24	0.41	1.79	1.71	2.38
Poisson	a = 2.0 a = 3.0 a = 4.0	-19 -24 -31	-6 -6 -6	0.44 0.32 0.25	1.30 1.85 2.08	1.21 1.15 1.75	1.69 2.08 2.58
Hanning- poisson	a=0.5 a=1.0 a=2.0	-35 -39 NONE	-18 -18 -18	0.43 0.38 0.29	1.61 1.73 2.02	1.54 1.64 a.87	2.14 2.30 2.65
Cauchy	a=3.0 a=4.0 a=5.0	-31 -35 -30	-6 -6 -6	0.42 0.33 0.28	1.48 1.76 2.06	1.34 1.50 1.68	1.90 2.20 2.53
Gaussian	a=2.5 a=3.0 a=3.5	-42 -55 -69	-6 -6 -6	0.51 0.43 0.37	1.39 1.64 1.90	1.33 1.55 1.79	1.86 2.18 2.52
Dolph- Chebyshev	a=2.5 a=3.0 a=3.5 a=4.0	-50 -60 -70 -80	0 0 0 0	0.53 0.48 0.45 0.42	1.39 1.51 1.62 1.73	1.33 1.44 1.55 1.65	1.85 2.01 2.17 2.31
Kaisser- Bessel	a=2.0 a=2.5 a=3.0 a=3.5	-46 -57 -69 -82	-6 -6 -6 -6	0.49 0.44 0.40 0.37	1.50 1.65 1.80 1.93	1.43 1.57 1.71 1.83	1.99 2.20 2.39 2.57

Eksterne henvisninger

• National Instruments: Windowing: Optimizing FFTs Using Window Functions Citat: "...Figure 8. Recommendations for different window types..."

Noter

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