The commonly used windows presented in Section 7.5.1 can all be expressed in terms of re...

The commonly used windows presented in Section 7.5.1 can all be expressed in terms of rectangular windows. This fact can be used to obtain expressions for the Fourier transforms of the Bartlett window and the raised-cosine family of windows, which includes the Hanning, Hamming, and Blackman windows.

(a) Show that the (M+1)-point Bartlett window, defined by Eq. (7.60b), can be expressed as the convolution of two smaller rectangular windows. Use this fact to show that the Fourier transform of the (M + 1)-point Bartlett window is

Or

(b) It can easily be seen that the (M+1)-point raised-cosine windows defined by Eqs. (7.60c)– (7.60e) can all be expressed in the form

where w_R[n] is an (M + 1)-point rectangular window. Use this relation to find the Fourier transform of the general raised-cosine window.

(c) Using appropriate choices for A, B, and C and the result determined in part (b), sketch the magnitude of the Fourier transform of the Hanning window.