In this problem we are interested in estimating the spectra of three very long real-valu...

In this problem we are interested in estimating the spectra of three very long real-valued data sequences x₁[n], x₂[n], and x₃[n], each consisting of the sum of two sinusoidal components. However, we only have a 256-point segment of each sequence available for analysis.

Let denote the 256-point segments of x₁[n], x₂[n], and x₃[n], respectively. We have some information about the nature of the spectra of the infinitely long sequences, as indicated in Eqs. (P10.22-1) through (P10.22-3). Two different spectral analysis procedures are being considered for use, one using a 256-point rectangular window and the other a 256-point Hamming window. These procedures are described below. In the descriptions, the signal R_N[n] denotes the N-point rectangular window and H_N[n] denotes the N-point Hamming window. The operator DFT₂₀₄₈{·} indicates taking the 2048-point DFT of its argument after zero-padding the end of the input sequence. This will give a good interpolation of the DTFT from the frequency samples of the DFT.

Based on Eqs. (P10.22-1) through (P10.22-3), indicate which of the spectral analysis procedures described below would allow you to conclude responsibly whether the anticipated frequency components were present. A good justification at a minimum will include a quantitative consideration of both resolution and side-lobe behavior of the estimators. Note that it is possible that both or neither of the algorithms will work for any given data sequence. Table 7.2 may be useful in deciding which algorithm(s) to use with which sequence.