The symmetric Bartlett window, which arises in many aspects of power spectrum estimation...

The symmetric Bartlett window, which arises in many aspects of power spectrum estimation, is defined as

The Bartlett window is particularly attractive for obtaining estimates of the power spectrum by windowing an estimated autocorrelation function, as discussed in Section 10.6. This is because its Fourier transform is nonnegative, which guarantees that the smoothed spectrum estimate will be nonnegative at all frequencies.

(a) Show that the Bartlett window as defined in Eq. (P10.35-1) is (1/M) times the aperiodic autocorrelation of the sequence (u[n] − u[n −M]).

(b) From the result of part (a), show that the Fourier transform of the Bartlett window is

which is clearly nonnegative.

(c) Describe a procedure for generating other finite-length window sequences that have nonnegative Fourier transforms.