In both digital filter design and spectral estimation, the choice of a windowing function can play an important role in determining the quality of overall results. The main role of the window is to damp out the effects of the Gibbs phenomenon that results from truncation of an infinite series.
Flat Top window
Nuttall's Blackman-Harris window
Parzen (de la Vallée-Poussin) window
Tapered cosine window
Two graphical user interface tools are provided for working with windows in the Signal Processing Toolbox™ product:
Refer to the reference pages for detailed information.
The basic window is the rectangular window, a vector of ones of the appropriate length. A rectangular window of length 50 is
n = 50; w = rectwin(n);
Signal Processing Toolbox stores windows in column vectors by convention, so an equivalent expression is
w = ones(50,1);
To use the Window Designer app to create this window, type
The app opens with a default Hamming window. To visualize the rectangular window, set Type = Rectangular and Length = 50 in the Current Window Information panel and then press Apply.
The Bartlett (or triangular) window is
the convolution of two rectangular windows. The functions
triang compute similar triangular windows, with three important
bartlett function always returns a window with
two zeros on the ends of the sequence, so that for
n odd, the
center section of
bartlett(n+2) is equivalent to
Bartlett = bartlett(7); isequal(Bartlett(2:end-1),triang(5))
ans = 1
bartlett is still the
convolution of two rectangular sequences. There is no standard definition for the
triangular window for
n even; the slopes of the line segments of
triang result are slightly steeper than those of
bartlett in this case:
w = bartlett(8); [w(2:7) triang(6)]
You can see the difference between odd and even Bartlett windows in Window Designer.
The final difference between the Bartlett and triangular windows is evident in the
Fourier transforms of these functions. The Fourier transform of a Bartlett window is
n even. The Fourier transform of a triangular
window, however, is always nonnegative.
The following figure, which plots the zero-phase responses of 8-point Bartlett and Triangular windows, illustrates the difference.
zerophase(bartlett(8)) hold on zerophase(triang(8)) legend('Bartlett','Triangular') axis([0.3 1 -0.2 0.5])
This difference can be important when choosing a window for some spectral estimation techniques, such as the Blackman-Tukey method. Blackman-Tukey forms the spectral estimate by calculating the Fourier transform of the autocorrelation sequence. The resulting estimate might be negative at some frequencies if the window's Fourier transform is negative.