Window Sequences

Spectral Refinement

Window Sequences

Some text about window sequences

Rectangular Windows

L      = 65;
h_rect = ones(L,1);

The most obvious window sequence is the so-called rectangular window, which means that all coefficients within a certain range are one and everything outside is blocked (meaning to be weighted with zero):

\begin{equation} w_{\textrm{rect},i} = \left\{ \begin{array}{ll} 1, & \textrm{for}\,\,\,i\in\Big\{-\frac{N}{2},\, ...,\, \frac{N}{2}\Big\}, \\ 0, & \textrm{else}. \end{array} \right. \end{equation}

In the equation above we have centered the window around index zero, which leads on the one hand side to a zero phase filter that has non-causal parts. On the other hand, we use this way of depticting the windows in order to be able to compare them in a simple way. Adding a delay of \(N/2\) samples will create a causal filter. In Matlab a (causal) rectangular window of length \(L=N+1\) can be created as described within the codebox. The following two pictures show the window sequence in the time domain as well as by means of the logarithmic magntiude in the spectral domain.

Triangle or Bartlett Windows

Cos Windows

Cos² or Hann Windows

Another \(w(n)\) set of window sequences is based on trigonoetric functions. One prominent example is the Hann window, which is specified as follows:

\begin{equation} w_{\textrm{han},i} = \left\{ \begin{array}{ll} \displaystyle{\frac{1}{2} - \frac{1}{2} \cos\left(\frac{2\pi}{N}i\right)}, & \textrm{for}\,\,\,i\in\{0,\, ...,\, N-1\}, \\ 0, & \textrm{else}. \end{array} \right. \end{equation}

The Hann window has its name from the Austian metrorogiest Julius von Hann. Some books use also the name "Hanning window" in order to be phonetically close to the Hamming window (according to the American mathematican Richard Hamming), but if one wants to stay close with the name of its inventor, one should use the name Hann. Julius Ferdinand von Hann was born in 1839 in a small city near Linz in Austria and died in 1921. According to Wikipidia he is seen as "a father of modern meteorology".