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| − | <math>X(\omega | + | <math>X(\omega) = F(x(t))</math> |
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| + | <math>X(\omega) = F(\cos( 4t + \frac{4\pi}{3}))</math> | ||
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| + | <math>X(\omega) = F(</math> | ||
Revision as of 09:10, 24 October 2008
Contents
- 1 Partial Fractions
- 2 The Four Cases to take into Account
- 2.1 Case 1: Denominator is a product of distinct linear factors.
- 2.2 Case 2: Denominator is a product of linear factors, some of which are repeated.
- 2.3 Case 3: Denominator contains irreducible quadratic factors, none of which is repeated.
- 2.4 Case 4: Denominator contains a repeated irreducible quadratic factor.
- 3 F.T. Logic
- 4 Example of a F.T.
Partial Fractions
This page is meant as a comprehensive review of partial fraction expansion. Partial fraction expansion allows us to fit functions to the known ones given by the known Fourier Transform pairs table.
First, the denominator must be of a higher degree than the numerator. If this is not the case, then perform long division to make it such. Note: for the remainder of this guide it is assumed that the denominator is of a higher degree than the numerator.
The Four Cases to take into Account
Case 1: Denominator is a product of distinct linear factors.
$ \frac{(Polynomial)}{(a_1x + b_1)(a_2x + b_2)...(a_kx + b_k)} = \frac{A_1}{(a_1x + b_1)}+\frac{A_2}{(a_2x + b_2)}+...+\frac{A_k}{(a_kx + b_k)} $
Case 2: Denominator is a product of linear factors, some of which are repeated.
$ \frac{(Polynomial)}{(a_1x + b_1)^r} = \frac{A_1}{a_1x + b_1}+\frac{A_2}{(a_1x + b_1)^2}+...+\frac{A_r}{(a_1x + b_1)^r} $
Case 3: Denominator contains irreducible quadratic factors, none of which is repeated.
$ \frac{(Polynomial)}{ax^x + bx + c} = \frac{Ax + B}{ax^+bx+c} $
Case 4: Denominator contains a repeated irreducible quadratic factor.
$ \frac{(Polynomial)}{(ax^x + bx + c)^r} = \frac{A_1x + B_2}{ax^+bx+c}+\frac{A_2x + B_2}{(ax^+bx+c)^2}+...+\frac{A_rx + B_r}{(ax^+bx+c)^r} $
F.T. Logic
Is the Signal Periodic?
Yes No
Is it in the table? Can you Integrate?
Yes No Yes No
Cool Try to break it up and Cool Is it in the table?
check the table again Yes No
Cool Try to break it up and reevaluate
Example of a F.T.
$ x(t) = \cos( 4t + \frac{4\pi}{3}) $
$ X(\omega) = F(x(t)) $
$ X(\omega) = F(\cos( 4t + \frac{4\pi}{3})) $
$ X(\omega) = F( $
