An analog signal consisting of a sum of sinusoids was sampled with a sampling rate of fs...

An analog signal consisting of a sum of sinusoids was sampled with a sampling rate of f_s = 10000 samples/s to obtain x[n] = x_c(nT ). Four spectrograms showing the time dependent Fourier transform |X[n, λ)| were computed using either a rectangular or a Hamming window. They are plotted in Figure P10.32. (A log amplitude scale is used, and only the top 35 dB is shown.)

(a) Which spectrograms were computed with a rectangular window?

(a) (b) (c) (d)

(b) Which pair (or pairs) of spectrograms have approximately the same frequency resolution?

(a&b) (b&d) (c&d) (a&d) (b&c)

(c) Which spectrogram has the shortest time window? (a) (b) (c) (d)

(d) To the nearest 100 samples, estimate the window length L (in samples) of the window in spectrogram (b).

(e) Use the spectrographic data in Figure P10.32 to assist you in writing an equation (or equations) for an analog sum of sinusoids x_c(t), which when sampled at a sampling rate of f_s = 10000, would produce the above spectrograms. Be as complete as you can in your description of the signal. Indicate any parameters that cannot be obtained from the spectrogram.