Please consider registering

sp_LogInOut Log In sp_Registration Register

Register | Lost password?
Advanced Search

— Forum Scope —

— Match —

— Forum Options —

Minimum search word length is 3 characters - maximum search word length is 84 characters

sp_Feed sp_TopicIcon
Algebra Question from Chapter 5 page 133 equation 5.56
No permission to create posts
January 3, 2015
1:34 am
Forum Posts: 8
Member Since:
December 25, 2014
sp_UserOfflineSmall Offline

Hey Will,

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.

Thank you

January 4, 2015
8:38 pm
W Pirkle
Forum Posts: 247
Member Since:
January 29, 2017
sp_UserOfflineSmall Offline

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.

January 6, 2015
1:34 am
Forum Posts: 8
Member Since:
December 25, 2014
sp_UserOfflineSmall Offline

Thank you so much for the clarification. I did not get the part how the a0 coefficient was eliminated. And your explanation for the second part that was a great deal of help as well.

Forum Timezone: America/New_York

Most Users Ever Online: 36

Currently Online:
4 Guest(s)

Currently Browsing this Page:
1 Guest(s)

Top Posters:

Skyler: 48

Derek: 46

Peter: 41

Frodson: 40

clau_ste: 39

Gwen: 32

JimmyM: 30

TheSmile: 27

EZB: 24

lppier: 23

Member Stats:

Guest Posters: 1

Members: 522

Moderators: 1

Admins: 4

Forum Stats:

Groups: 11

Forums: 31

Topics: 545

Posts: 2134

Newest Members:

Bill, hill william, NAUN_SONAR, sufy, Diane, Richard, drvenkman, venkman, Jfx, drorh

Moderators: W Pirkle: 247

Administrators: Tom: 67, JD Young: 80, Will Pirkle: 0, W Pirkle: 247