This problem is about designing LSI graph filters using least squares. An LSI graph filter can be defined through its frequency response h ( ? `) at its distinct frequencies ? `, for ` = 0 , 1, . . . , N ? 1. In the weight matrix based DSP G framework, an LSI graph filter is a polynomial (in the weight matrix) of degree M. To construct an LSI filter with frequency response h ( ? `) = b `, one need to solve a system of N linear equations with M + 1 unknowns h 0, h 1, . . . , h M
In general, N > M + 1, and therefore, the system of linear equations is overdetermined
and does not have a solution. In this case, an approximate solution can be found in the
least-squares sense.
Design the filters with the following specifications using least squares for the graph shown in
Figure 10.7.
(a) A low-pass filter of order two that passes only the smallest two frequencies.
(b) A high-pass filter of order two that passes only the largest two frequencies.
(c) A band-pass filter of order two that passes only two frequency components ?2 and ?3.
(a) The filter passes only the smallest two frequencies (?0 and?1), therefore, assumingb0=b1 = 1 andb2=b3=b4 = 0.
The filter parameters can be computed using Equation (10.11) as
which is plotted in Figure 10.8. Note that the filter designed using least squares method is
not ideal, because we used only three number of parameters to design the filter. To design
the ideal filter, we will need five number of parameters. However, in the designed filter using
least squares, the filter gain for the smallest two frequencies is much higher than the gain at
rest of the frequencies, which is a reasonable approximation.
(b) The filter passes only the highest two frequencies (?3 and?4), therefore, assumingb0=b1=b2 = 0 andb3=b4 = 1. The filter parameters can be computed using Equation (10.11) as
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