December 25, 2014
I have been going through your book and I am having a great time reading through it. I did not know where to place this topic so if it needs to be moved please do so.
I want to know how you derived equation 5.56 from equation 5.55. I understand that you multiplied complex conjugates to have the imaginary components disappear, but I cannot recall how to do so. I want to see the steps you took to get to 5.56 if possible.
January 28, 2017
The steps are more logical than symbolic-math. First, obviously you let a0 = 1.0 since it is just a scaling factor. Then, you wind up with an algebra equation not unlike x^2 + x + 1, which you factor into its parts. In algebra class, you are given pure real equations like
ax^2 + bx + 1 = 0
and then you factor to find the roots. The generalized solution is
(x - C)(x - D) = 0 and it is up to you to figure out C and D. You probably did this by trial and error back in algebra class (high-school as we call it in the states).
But, for a complex equation H(z), you still decompose the equation as
(x - C)(x - D)
Where C and D are complex values with real and imaginary parts. Since after multiplication, the final equation has only real coefficients (marked as a and b in the simple equation above) then C and D must be complex conjugates of each other, or D = C*
C = a + jb
C* = a - jb
When you multiply, the imaginary component disappears. To do this by hand, use the old FOIL rule (first, inside, outside, last) on (x - C)(x - C*)
x^2 - Cx - C*x + (C)(C*) and substitute a+jb and a-jb above.
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