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I don't understand how the graph of H(w) is used to find the equation for H(150) and H(200.)

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For the values of ω=100 to ω=300, |H(ω)| is a straight line and you have

x = ω = 100, y = |H(100)| = 16

x = ω = 300, y = |H(300)| = 24

So, solve the equation y = kx + c with the aforementioned values:

24 = k300 + c and 16 = k100 + c

And you get k = 4/100 and c = 12. So,

|H(150)| = (4/100) × 150 + 12 = 18

|H(200)| = (4/100) × 200 + 12 = 20

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If H was simply a filter its values would lie less than or equal to one, this means that there is gain there. The gain is linearly increasing from 100 to 300 (units of w) so the H(150) and H(200) is simply a calculation to get the values at those values of W. To check this out, simply realize that H(200) is exactly half way between H(100) and H(300). 20 is half way between 24 and 16.

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