Kaiser window

The Kaiser window and its Fourier transform are given by:

:$$

w_0(x) \triangleq \left\{

\begin{array}{ccl}

\tfrac{1}{L}\frac{I_0\left[\pi\alpha \sqrt{1 - \left(2x/L\right)^2}\right]}{I_0[\pi\alpha]},\quad &\left|x\right| \leq L/2\\

0,\quad &\left|x\right| > L/2

\end{array}\right\}

\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad

\frac{\sin\bigg(\sqrt{(\pi L f)^2-(\pi \alpha)^2}\bigg)}

{I_0(\pi \alpha)\cdot \sqrt{(\pi L f)^2-(\pi \alpha)^2}},

  {{cite journal

| doi =10.1109/TASSP.1981.1163506

| last =Nuttall

| first =Albert H.

| title =Some Windows with Very Good Sidelobe Behavior

| journal =IEEE Transactions on Acoustics, Speech, and Signal Processing

| volume =29

| issue =1

| page =89 (eq.38)

| date =Feb 1981

| url =https://zenodo.org/record/1280930

}}{{efn-ua

|An equivalent formula is:

{{cite web| url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_Window.html | website=ccrma.stanford.edu | title=Kaiser Window in Spectral Audio Signal Processing, eq.(4.40 & 4.42) | last=Smith | first=J.O. | date=2011 | access-date=2022-01-01}}

where $\backslash beta\; \backslash triangleq\; \backslash pi\; \backslash alpha,\backslash \; \backslash omega\; \backslash triangleq\; 2\; \backslash pi\; f,\backslash \; M=L.$

:$\backslash frac\{\backslash sinh\backslash bigg(\backslash sqrt\{(\backslash pi\; \backslash alpha)^2\; -\; (\backslash pi\; L\; f)^2\}\backslash bigg)\}$

{I_0(\pi \alpha)\cdot \sqrt{(\pi \alpha)^2 - (\pi L f)^2}}.

}}

Image:Kaiser-Window-Spectra.svg

where:

• {{math|I₀}} is the zeroth-order modified Bessel function of the first kind,
• {{mvar|L}} is the window duration, and
• {{math|α}} is a non-negative real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design.
• Sometimes the Kaiser window is parametrized by {{math|β}}, where {{math|β {{=}} πα}}.

For digital signal processing, the function can be sampled symmetrically as:

:$w[n]\; =\; L\backslash cdot\; w\_0\backslash left(\backslash tfrac\{L\}\{N\}\; (n-N/2)\backslash right)\; =\; \backslash frac\{I\_0\backslash left[\backslash pi\backslash alpha\; \backslash sqrt\{1\; -\; \backslash left(\backslash frac\{2n\}\{N\}-1\backslash right)^2\}\backslash right]\}\{I\_0[\backslash pi\backslash alpha]\},\backslash quad\; 0\; \backslash leq\; n\; \backslash leq\; N,$

where the length of the window is $N+1,$ and N can be even or odd. (see A list of window functions)

In the Fourier transform, the first null after the main lobe occurs at $f\; =\; \backslash tfrac\{\backslash sqrt\{1+\backslash alpha^2\}\}\{L\},$ which is just $\backslash sqrt\{1+\backslash alpha^2\}$ in units of N (DFT "bins"). As α increases, the main lobe increases in width, and the side lobes decrease in amplitude.  {{math|α}} = 0 corresponds to a rectangular window. For large {{math|α,}} the shape of the Kaiser window (in both time and frequency domain) tends to a Gaussian curve.  The Kaiser window is nearly optimal in the sense of its peak's concentration around frequency $0.$

{{Cite book

|last=Oppenheim

|first=Alan V.

|authorlink=Alan V. Oppenheim

|last2=Schafer

|first2=Ronald W.

|author2-link=Ronald W. Schafer

|last3=Buck

|first3=John R.

|title=Discrete-time signal processing

|year=1999

|publisher=Prentice Hall

|location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

|chapter=7.2

|page=[https://archive.org/details/discretetimesign00alan/page/474 474]

|quote=a near-optimal window could be formed using the zeroth-order modified Bessel function of the first kind

|url-access=registration

|url=https://archive.org/details/discretetimesign00alan

}}

{{clear}}

File:Kbd-window.svg

A related window function is the Kaiser–Bessel-derived (KBD) window, which is designed to be suitable for use with the modified discrete cosine transform (MDCT). The KBD window function is defined in terms of the Kaiser window of length N+1, by the formula:

:$$

d_n =

\begin{cases}

\sqrt{\frac{\sum_{i=0}^{n} w[i]} {\sum_{i=0}^N w[i]}}

& \mbox{if } 0 \leq n < N \\

\sqrt{\frac{\sum_{i=0}^{2N-1-n} w[i]} {\sum_{i=0}^N w[i]}}

& \mbox{if } N \leq n \leq 2N-1 \\

0 & \mbox{otherwise}. \\

\end{cases}

This defines a window of length 2N, where by construction d_n satisfies the Princen-Bradley condition for the MDCT (using the fact that {{math|w{{sub|N−n}} {{=}} w{{sub|n}}}}): {{math|d{{sub|n}}² + (d{{sub|n+N}})² {{=}} 1}} (interpreting n and n + N modulo 2N). The KBD window is also symmetric in the proper manner for the MDCT: d_n = d_2N−1−n.

The KBD window is used in the Advanced Audio Coding digital audio format.

{{notelist-ua}}

• {{cite journal

|doi=10.1109/PROC.1978.10837

|last=Harris

|first=Fredric J.

|title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform

|journal=Proceedings of the IEEE

|volume=66

|issue=1

|page=73 (eq 46b)

|date=Jan 1978

|url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880

}}

• {{Cite journal |last1=Kaiser |first1=James F. |last2=Schafer |first2=Ronald W. |doi=10.1109/TASSP.1980.1163349 |title=On the use of the I₀-sinh window for spectrum analysis |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=28 |pages=105–107 |year=1980}}
• {{Cite web|url=https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html|website=ccrma.stanford.edu| title =Spectral Audio Signal Processing, Kaiser and DPSS Windows Compared | last =Smith | first =J.O. | date =2011 |access-date=2016-04-13}}
• {{cite web| url =https://www.mathworks.com/help/signal/ug/kaiser-window.html | website=www.mathworks.com | title =Kaiser Window, R2018b | publisher =Mathworks | access-date =2019-03-20}}

Category:Digital_signal processing