Signal Processing Toolbox

Basic Shapes

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 = boxcar(n);

This toolbox stores windows in column vectors by convention, so an equivalent expression is

w = ones(50,1);

The Bartlett (or triangular) window is the convolution of two rectangular windows. The functions bartlett and triang compute similar triangular windows, with three important differences. The 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 triang(n).

bartlett(7)
ans =
         0
    0.3333
    0.6667
    1.0000
    0.6667
    0.3333
         0
triang(5)
ans =
    0.3333
    0.6667
    1.0000
    0.6667
    0.3333

For n even, 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 the triang result are slightly steeper than those of bartlett in this case.

w = bartlett(8); 
[w(2:7)  triang(6)]
ans =
    0.2857    0.1667
    0.5714    0.5000
    0.8571    0.8333
    0.8571    0.8333
    0.5714    0.5000
    0.2857    0.1667

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 negative for n even. The Fourier transform of a triangular window, however, is always nonnegative.

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 (see Kay [1], pg. 80).

Windows

Generalized Cosine Windows