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Hann function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Hamming_function" class="mw-redirect" title="Hamming function">Hamming function</a>.</div>

The Hann function is named after the Austrian meteorologist <a href="Julius_von_Hann" title="Julius von Hann">Julius von Hann</a>. It is a <a href="Window_function" title="Window function">window function</a> used to perform Hann smoothing or hanning.<a href="#cite_note-Essenwanger-1">[1]</a><a href="#cite_note-Kahlig-2">[2]</a> The function, with length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> and amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/L,}">

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<annotation encoding="application/x-tex">{\displaystyle 1/L,}</annotation>

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</math><img src="./b989a3de7994ab6e0e0ba4c22fc9b7ab01a1e3c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle 1/L,}" loading="lazy"> is given by:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\cos \left({\frac {2\pi x}{L}}\right)\right)={\tfrac {1}{L}}\cos ^{2}\left({\frac {\pi x}{L}}\right),\quad &\left|x\right|\leq L/2\\0,\quad &\left|x\right|>L/2\end{array}}\right\}.}">

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\cos \left({\frac {2\pi x}{L}}\right)\right)={\tfrac {1}{L}}\cos ^{2}\left({\frac {\pi x}{L}}\right),\quad &\left|x\right|\leq L/2\\0,\quad &\left|x\right|>L/2\end{array}}\right\}.}</annotation>

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</math><img src="./3a22873e0ddd8a71818adf45bc8e68c7e18d6cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:65.598ex; height:8.176ex;" alt="{\displaystyle w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{L}}\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\cos \left({\frac {2\pi x}{L}}\right)\right)={\tfrac {1}{L}}\cos ^{2}\left({\frac {\pi x}{L}}\right),\quad &\left|x\right|\leq L/2\\0,\quad &\left|x\right|>L/2\end{array}}\right\}.}" loading="lazy">   <a href="#cite_note-3">[a]</a></dd></dl>

For <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, the function is sampled symmetrically (with spacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/N}">

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<annotation encoding="application/x-tex">{\displaystyle \left.{\begin{aligned}w[n]=L\cdot w_{0}\left({\tfrac {L}{N}}(n-N/2)\right)&={\tfrac {1}{2}}\left[1-\cos \left({\tfrac {2\pi n}{N}}\right)\right]\\&=\sin ^{2}\left({\tfrac {\pi n}{N}}\right)\end{aligned}}\right\},\quad 0\leq n\leq N,}</annotation>

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</math><img src="./83b723018aa0923be6327c07fed4753f1ed7d033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:68.162ex; height:9.509ex;" alt="{\displaystyle \left.{\begin{aligned}w[n]=L\cdot w_{0}\left({\tfrac {L}{N}}(n-N/2)\right)&={\tfrac {1}{2}}\left[1-\cos \left({\tfrac {2\pi n}{N}}\right)\right]\\&=\sin ^{2}\left({\tfrac {\pi n}{N}}\right)\end{aligned}}\right\},\quad 0\leq n\leq N,}" loading="lazy"></dd></dl>

which is a sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1}">

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</math><img src="./fdf2b9cbfe9051fd4e7b50c8028866d497eac35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N+1}" loading="lazy"> samples, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> can be even or odd. It is also known as the raised cosine window, Hann filter, von Hann window, Hanning window, etc.<a href="#cite_note-Kahlig-2">[2]</a><a href="#cite_note-Smith-4">[3]</a><a href="#cite_note-Blackman-5">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Fourier_transform">Fourier transform</h2></div>

The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}(f)={\frac {1}{2}}{\frac {\operatorname {sinc} (Lf)}{(1-L^{2}f^{2})}}={\frac {\sin(\pi Lf)}{2\pi Lf(1-L^{2}f^{2})}}}</annotation>

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</math><img src="./2f83f6582887bbe5f9e22013857fdc5274209182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.207ex; height:6.509ex;" alt="{\displaystyle W_{0}(f)={\frac {1}{2}}{\frac {\operatorname {sinc} (Lf)}{(1-L^{2}f^{2})}}={\frac {\sin(\pi Lf)}{2\pi Lf(1-L^{2}f^{2})}}}" loading="lazy">   <a href="#cite_note-6">[b]</a></dd></dl>

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</style><div class="math_proof" style="">Derivation

Using <a href="Euler's_formula" title="Euler's formula">Euler's formula</a> to expand the cosine term in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{0}(x),}">

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<annotation encoding="application/x-tex">{\displaystyle w_{0}(x)={\tfrac {1}{L}}\left({\tfrac {1}{2}}\operatorname {rect} (x/L)+{\tfrac {1}{4}}e^{i2\pi x/L}\operatorname {rect} (x/L)+{\tfrac {1}{4}}e^{-i2\pi x/L}\operatorname {rect} (x/L)\right),}</annotation>

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</math><img src="./f64c9e175034eb14423a1af18f8c2b04fb1a7f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:70.556ex; height:4.843ex;" alt="{\displaystyle w_{0}(x)={\tfrac {1}{L}}\left({\tfrac {1}{2}}\operatorname {rect} (x/L)+{\tfrac {1}{4}}e^{i2\pi x/L}\operatorname {rect} (x/L)+{\tfrac {1}{4}}e^{-i2\pi x/L}\operatorname {rect} (x/L)\right),}" loading="lazy"></dd></dl>

which is a linear combination of modulated <a href="Rectangular_function" title="Rectangular function">rectangular windows</a>:

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<mtext>Fourier transform</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{L}}\operatorname {rect} (x/L)\quad {\stackrel {\text{Fourier transform}}{\longleftrightarrow }}\quad \operatorname {sinc} (Lf)\triangleq {\frac {\sin(\pi Lf)}{\pi Lf}}.}</annotation>

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</math><img src="./62b6603868b7280213b2ea26e59797ad541127ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.012ex; height:6.176ex;" alt="{\displaystyle {\tfrac {1}{L}}\operatorname {rect} (x/L)\quad {\stackrel {\text{Fourier transform}}{\longleftrightarrow }}\quad \operatorname {sinc} (Lf)\triangleq {\frac {\sin(\pi Lf)}{\pi Lf}}.}" loading="lazy"></dd></dl>

Transforming each term:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}W_{0}(f)&={\tfrac {1}{2}}\operatorname {sinc} (Lf)+{\tfrac {1}{4}}\operatorname {sinc} (L(f-1/L))+{\tfrac {1}{4}}\operatorname {sinc} (L(f+1/L))\\&={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi Lf}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf-1))}{\pi (Lf-1)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf+1))}{\pi (Lf+1)}}\\&={\frac {1}{2\pi }}\left({\frac {\sin(\pi Lf)}{Lf}}-{\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{Lf-1}}-{\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{Lf+1}}\right)\\&={\frac {\sin(\pi Lf)}{2\pi }}\left({\frac {1}{Lf}}+{\tfrac {1}{2}}{\frac {1}{1-Lf}}-{\tfrac {1}{2}}{\frac {1}{1+Lf}}\right)\\&={\frac {\sin(\pi Lf)}{2\pi }}\cdot {\frac {1}{Lf(1-Lf)(1+Lf)}}={\frac {1}{2}}{\frac {\operatorname {sinc} (Lf)}{(1-L^{2}f^{2})}}.\end{aligned}}}</annotation>

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</math><img src="./8c2bee3303cf6404c1bb416351194140f8e18249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:64.793ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}W_{0}(f)&={\tfrac {1}{2}}\operatorname {sinc} (Lf)+{\tfrac {1}{4}}\operatorname {sinc} (L(f-1/L))+{\tfrac {1}{4}}\operatorname {sinc} (L(f+1/L))\\&={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi Lf}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf-1))}{\pi (Lf-1)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf+1))}{\pi (Lf+1)}}\\&={\frac {1}{2\pi }}\left({\frac {\sin(\pi Lf)}{Lf}}-{\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{Lf-1}}-{\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{Lf+1}}\right)\\&={\frac {\sin(\pi Lf)}{2\pi }}\left({\frac {1}{Lf}}+{\tfrac {1}{2}}{\frac {1}{1-Lf}}-{\tfrac {1}{2}}{\frac {1}{1+Lf}}\right)\\&={\frac {\sin(\pi Lf)}{2\pi }}\cdot {\frac {1}{Lf(1-Lf)(1+Lf)}}={\frac {1}{2}}{\frac {\operatorname {sinc} (Lf)}{(1-L^{2}f^{2})}}.\end{aligned}}}" loading="lazy"></dd></dl>

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<div class="mw-heading mw-heading2"><h2 id="Discrete_transforms">Discrete transforms</h2></div>

The <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT) of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1}">

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</math><img src="./fdf2b9cbfe9051fd4e7b50c8028866d497eac35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N+1}" loading="lazy"> length, time-shifted sequence is defined by a Fourier series, which also has a 3-term equivalent that is derived similarly to the Fourier transform derivation:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\{w[n]\}&\triangleq \sum _{n=0}^{N}w[n]\cdot e^{-i2\pi fn}\\&=e^{-i\pi fN}\left[{\tfrac {1}{2}}{\frac {\sin(\pi (N+1)f)}{\sin(\pi f)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (N+1)(f-{\tfrac {1}{N}}))}{\sin(\pi (f-{\tfrac {1}{N}}))}}+{\tfrac {1}{4}}{\frac {\sin(\pi (N+1)(f+{\tfrac {1}{N}}))}{\sin(\pi (f+{\tfrac {1}{N}}))}}\right].\end{aligned}}}</annotation>

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</math><img src="./833f432fd5ae9ae88eb7ef4247952f92c7aef4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:95.56ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}\{w[n]\}&\triangleq \sum _{n=0}^{N}w[n]\cdot e^{-i2\pi fn}\\&=e^{-i\pi fN}\left[{\tfrac {1}{2}}{\frac {\sin(\pi (N+1)f)}{\sin(\pi f)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (N+1)(f-{\tfrac {1}{N}}))}{\sin(\pi (f-{\tfrac {1}{N}}))}}+{\tfrac {1}{4}}{\frac {\sin(\pi (N+1)(f+{\tfrac {1}{N}}))}{\sin(\pi (f+{\tfrac {1}{N}}))}}\right].\end{aligned}}}" loading="lazy"></dd></dl>

The truncated sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{w[n],\ 0\leq n\leq N-1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{w[n],\ 0\leq n\leq N-1\}}</annotation>

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</math><img src="./bd1c0e1f3c303a9cac460a10ca4b6e00339befb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.113ex; height:2.843ex;" alt="{\displaystyle \{w[n],\ 0\leq n\leq N-1\}}" loading="lazy"> is a <a href="Spectral_leakage#DFT-symmetry" title="Spectral leakage">DFT-even</a> (aka periodic) Hann window. Since the truncated sample has value zero, it is clear from the Fourier series definition that the DTFTs are equivalent. However, the approach followed above results in a significantly different-looking, but equivalent, 3-term expression:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{w[n]\}=e^{-i\pi f(N-1)}\left[{\tfrac {1}{2}}{\frac {\sin(\pi Nf)}{\sin(\pi f)}}+{\tfrac {1}{4}}e^{-i\pi /N}{\frac {\sin(\pi N(f-{\tfrac {1}{N}}))}{\sin(\pi (f-{\tfrac {1}{N}}))}}+{\tfrac {1}{4}}e^{i\pi /N}{\frac {\sin(\pi N(f+{\tfrac {1}{N}}))}{\sin(\pi (f+{\tfrac {1}{N}}))}}\right].}</annotation>

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</math><img src="./366f2cccda229534d868deeaa79031fef7368df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:92.243ex; height:8.009ex;" alt="{\displaystyle {\mathcal {F}}\{w[n]\}=e^{-i\pi f(N-1)}\left[{\tfrac {1}{2}}{\frac {\sin(\pi Nf)}{\sin(\pi f)}}+{\tfrac {1}{4}}e^{-i\pi /N}{\frac {\sin(\pi N(f-{\tfrac {1}{N}}))}{\sin(\pi (f-{\tfrac {1}{N}}))}}+{\tfrac {1}{4}}e^{i\pi /N}{\frac {\sin(\pi N(f+{\tfrac {1}{N}}))}{\sin(\pi (f+{\tfrac {1}{N}}))}}\right].}" loading="lazy"></dd></dl>

An N-length DFT of the window function samples the DTFT at frequencies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=k/N,}">

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<annotation encoding="application/x-tex">{\displaystyle f=k/N,}</annotation>

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</math><img src="./a9a584500b8f983993803a822dfe527df8227a32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.461ex; height:2.843ex;" alt="{\displaystyle f=k/N,}" loading="lazy"> for integer values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k.}">

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<annotation encoding="application/x-tex">{\displaystyle k.}</annotation>

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</math><img src="./bcb6778a29f576eb23da1dbddffb73b2571359ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k.}" loading="lazy"> From the expression immediately above, it is easy to see that only 3 of the N DFT coefficients are non-zero. And from the other expression, it is apparent that all are real-valued. These properties are appealing 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 <a href="Discrete_Fourier_transform#Convolution_theorem_duality" title="Discrete Fourier transform">convolution</a>.<a href="#cite_note-Carlin-7">[5]</a><a href="#cite_note-8">[c]</a><a href="#cite_note-9">[d]</a>

The function is named in honor of von Hann, who used the three-term weighted average smoothing technique on meteorological data.<a href="#cite_note-Hann-10">[6]</a><a href="#cite_note-Kahlig-2">[2]</a> However, the term Hanning function is also conventionally used,<a href="#cite_note-Harris-11">[7]</a> derived from the paper in which the term hanning a signal was used to mean applying the Hann window to it.<a href="#cite_note-Blackman-5">[4]</a><a href="#cite_note-Blackman2-12">[8]</a> It is distinct from the similarly-named <a href="Hamming_function" class="mw-redirect" title="Hamming function">Hamming function</a>, named after <a href="Richard_Hamming" title="Richard Hamming">Richard Hamming</a>.

<ul><li><a href="Window_function" title="Window function">Window function</a></li>

<li><a href="Apodization" title="Apodization">Apodization</a></li>

<li><a href="Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine distribution</a></li>

<li><a href="Raised-cosine_filter" title="Raised-cosine filter">Raised-cosine filter</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Page_citations">Page citations</h2></div>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#Nuttall">Nuttall 1981</a>, p 84 (3)

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#Nuttall">Nuttall 1981</a>, p 86 (17)

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#Nuttall">Nuttall 1981</a>, p 85

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#Harris">Harris 1978</a>, p 62

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFEssenwanger,_O._M._(Oskar_M.)1986" class="citation book cs1">Essenwanger, O. M. (Oskar M.) (1986). Elements of statistical analysis. Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0444424261</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/152410575">152410575</a>.</cite>

</li>

<cite id="CITEREFKahlig1993" class="citation cs2">Kahlig, Peter (1993), <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/260824978">"Some aspects of Julius von Hann's contribution to modern climatology"</a>, in McBean, G.A.; Hantel, M. (eds.), Interactions Between Global Climate Subsystems: The Legacy of Hann, Geophysical Monograph Series, vol. 75, American Geophysical Union, pp. 1–7, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2Fgm075p0001">10.1029/gm075p0001</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780875904665</bdi>, retrieved 2019-07-01, <q>Hann appears to be the inventor of a certain data smoothing procedure, now called "hanning" ... or "Hann smoothing" ... Essentially, it is a three-term moving average (running mean) with unequal weights (1/4, 1/2, 1/4).</q></cite>

</li>

<cite id="CITEREFSmith,_Julius_O._(Julius_Orion)2011" class="citation book cs1">Smith, Julius O. (Julius Orion) (2011). <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/sasp/Hann_Hanning_Raised_Cosine.html">Spectral audio signal processing</a>. Stanford University. Center for Computer Research in Music and Acoustics., Stanford University. Department of Music. [Stanford, Calif.?]: W3K. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780974560731</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/776892709">776892709</a>.</cite>

</li>

<cite id="CITEREFBlackmanTukey1958" class="citation journal cs1"><a href="R._B._Blackman" class="mw-redirect" title="R. B. Blackman">Blackman, R. B.</a>; Tukey, J. W. (1958). "The measurement of power spectra from the point of view of communications engineering — Part I". The Bell System Technical Journal. 37 (1): 273. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1958.tb03874.x">10.1002/j.1538-7305.1958.tb03874.x</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0005-8580">0005-8580</a>.</cite>

</li>

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</style><a rel="nofollow" class="external text" href="https://worldwide.espacenet.com/textdoc?DB=EPODOC&IDX=US6898235">US patent 6898235</a>, Carlin, Joe; Collins, Terry & Hays, Peter et al., "Wideband communication intercept and direction finding device using hyperchannelization", published 1999-12-10, issued 2005-05-24<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Apatent&rft.number=6898235&rft.cc=US&rft.title=Wideband+communication+intercept+and+direction+finding+device+using+hyperchannelization&rft.inventor=Carlin%2C+Joe&rft.date=2005-05-24&rft.appldate=1999-12-10&rft.pubdate=1999-12-10"> ,

also available at <a rel="nofollow" class="external free" href="https://patentimages.storage.googleapis.com/4d/39/2a/cec2ae6f33c1e7/US6898235.pdf">https://patentimages.storage.googleapis.com/4d/39/2a/cec2ae6f33c1e7/US6898235.pdf</a>

</li>

<cite id="CITEREFvon_Hann1903" class="citation book cs1">von Hann, Julius (1903). <a rel="nofollow" class="external text" href="https://archive.org/details/handbookclimato01wardgoog">Handbook of Climatology</a>. Macmillan. p. <a rel="nofollow" class="external text" href="https://archive.org/details/handbookclimato01wardgoog/page/n219">199</a>. <q>The figures under b are determined by taking into account the parallels 5° away on either side. Thus, for example, for latitude 60° we have ½[60 + (65 + 55)÷2].</q></cite>

</li>

<cite id="Harris" class="citation journal cs1">Harris, Fredric J. (Jan 1978). <a rel="nofollow" class="external text" href="http://web.mit.edu/xiphmont/Public/windows.pdf">"On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform"</a> (PDF). Proceedings of the IEEE. 66 (1): 51–83. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.649.9880">10.1.1.649.9880</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPROC.1978.10837">10.1109/PROC.1978.10837</a>. <q>The correct name of this window is 'Hann.' The term 'Hanning' is used in this report to reflect conventional usage. The derived term 'Hann'd' is also widely used.</q></cite>

</li>

<cite id="CITEREFBlackmanTukey1959" class="citation book cs1"><a href="Ralph_Beebe_Blackman" title="Ralph Beebe Blackman">Blackman, R. B. (Ralph Beebe)</a>; Tukey, John W. (John Wilder) (1959). <a rel="nofollow" class="external text" href="https://archive.org/details/TheMeasurementOfPowerSpectra">The measurement of power spectra from the point of view of communications engineering</a>. New York : Dover Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/TheMeasurementOfPowerSpectra/page/n58">98</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/59-10185">59-10185</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)

</li>

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<ol><li value="9"><cite id="Nuttall" class="citation journal cs1">Nuttall, Albert H. (Feb 1981). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1280930">"Some Windows with Very Good Sidelobe Behavior"</a>. IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTASSP.1981.1163506">10.1109/TASSP.1981.1163506</a>.</cite></li></ol>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HanningFunction.html">Hann function</a> at <a href="MathWorld" title="MathWorld">MathWorld</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-23" href="https://en.wikipedia.org/wiki/?title=Hann_function&oldid=1291738506">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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