Hann function

The Hann function, named after the Austrian meteorologist Julius von Hann, is a window function given by:

w_{0}(x)\triangleq \left\{{\begin{array}{ccl}{\tfrac {1}{2}}\left(1+\cos \left({\frac {2\pi x}{L}}\right)\right)=\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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For digital signal processing, the function can be sampled symmetrically as:

\left.{\begin{aligned}w[n]=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,

where the length of the window is $N+1,$ and N can be even or odd. (see Window_function#A_list_of_window_functions)

Fourier transform

The Hann window is a linear combination of modulated rectangular windows:

\mathrm {rect} (x/L)\quad {\stackrel {\text{Fourier transform}}{\longleftrightarrow }}\quad L\cdot \mathrm {sinc} (Lf)={\frac {\sin(\pi Lf)}{\pi f}}.

Using Euler's formula to expand the cosine term, we can write:

w_{0}(x)={\tfrac {1}{2}}\mathrm {rect} (x/L)+{\tfrac {1}{4}}e^{i2\pi x/L}\mathrm {rect} (x/L)+{\tfrac {1}{4}}e^{-i2\pi x/L}\mathrm {rect} (x/L),

whose Fourier transform is just:

W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f-1/L))}{\pi (f-1/L)}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f+1/L))}{\pi (f+1/L)}}.

Discrete transforms

The Discrete-time Fourier transform (DTFT) of the N+1 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:

{\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}}

For even values of N, the truncated sequence $\{w[n],\ 0\leq n\leq N-1\}$ is a DFT-even (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:

{\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].

An N-length DFT of the window function samples the DTFT at frequencies $f=k/N,$ for integer values of $k.$ 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 convolution.^[2]

Name

Hann function is the original name, in honour of von Hann; however, the erroneous "Hanning" function is also heard of on occasion, derived from the paper in which it was named, where the term "hanning a signal" was used to mean applying the Hann window to it.^[3] The confusion arose from the similar Hamming function, named after Richard Hamming.

References

^ Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837.
^ US patent 6898235, Carlin, Joe; Collins, Terry & Hays, Peter et al., "Wideband communication intercept and direction finding device using hyperchannelization", issued 2005
^ Blackman, R. B. (Ralph Beebe); Tukey, John W. (John Wilder) (1959). The measurement of power spectra from the point of view of communications engineering. New York : Dover Publications. p. 98. LCCN 59-10185.{{cite book}}: CS1 maint: publisher location (link)

External links

Hann function at MathWorld

[1] Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837.

[2] US patent 6898235, Carlin, Joe; Collins, Terry & Hays, Peter et al., "Wideband communication intercept and direction finding device using hyperchannelization", issued 2005

[3] Blackman, R. B. (Ralph Beebe); Tukey, John W. (John Wilder) (1959). The measurement of power spectra from the point of view of communications engineering. New York : Dover Publications. p. 98. LCCN 59-10185.{{cite book}}: CS1 maint: publisher location (link)

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