mws_gen_fft_spe_ppttheoreticalfourier

mws_gen_fft_spe_ppttheoreticalfourier - Numerical Methods...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Numerical Methods Fast Fourier Transform Part: Theoretical Development of Fast Fourier Transform http://numericalmethods.eng.usf.edu ( ) 4 2 2 = = N
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
For more details on this topic Go to http://numericalmethods.eng.usf.edu Click on Keyword Click on Fast Fourier Transform
Background image of page 2
You are free to Share – to copy, distribute, display and perform the work to Remix – to make derivative works
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Under the following conditions Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial — You may not use this work for commercial purposes. Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.
Background image of page 4
Major: All Engineering Majors Authors: Duc Nguyen http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates http://numericalmethods.eng.usf.edu 9/4/2010 5 Chapter 11.06: Theoretical Development of Fast Fourier Transform Lecture # 16 ( ) 4 2 2 = = N
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Theoretical Development of FFT For the case of r N 2 = Recall Equation (5) from Chapter 11.05 Informal Development of FFT, = = = 1 0 ) ( ) ( ~ ~ N k nk n W k f n C C where N i e W π 2 = 9/4/2010 6 (11.65) Consider the case . In this case, we can express and as 2-bit binary numbers: k n 4 2 2 2 = = = r N ) 1 , 1 ( ), 0 , 1 ( ), 1 , 0 ( ), 0 , 0 ( ) , ( ) 3 , 2 , 1 , 0 ( 0 1 = = = k k k (1) (2)
Background image of page 6
Theoretical Development cont. Eqs. (1) and (2) can also be expressed in compact forms, as following 0 0 1 1 0 1 2 2 2 k k k k k + = + = 0 0 1 1 0 1 2 2 2 n n n n n + = + = 1 or , 0 , , , 0 1 0 1 = n n k k where (3) (4) In the new notations, Eq.(11.65) becomes ∑ ∑ = = + + = 1 0 0 1 0 1 ) 0 1 2 )( 0 1 2 ( 0 1 0 1 ) , ( ) , ( ~ k k k k n n W k k f n n C (5) 9/4/2010 7
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
http://numericalmethods.eng.usf.edu 8 Consider ( )( ) ( ) ( ) 0 0 1 2 1 2 0 1 2 0 1 2 0 1 2 * k n n k n n k k n n W W W + + + + = ( ) 0 0 1 2 1 0 2 1 1 4 k n n k n k n W W W + = ( ) 0 0 1 2 1 0 2 1 1 4 ] [ k n n k n k n W W W + = Theoretical Development cont.
Background image of page 8
Theoretical Development cont. Notice that [ ] 4 2 4 = N i e W π 4 4 2 = i e 1 ) 2 sin( ) 2 cos( 2 = = = i e i Hence Eq. (5) can be simplified to = + = = 1 0 0 0 ) 0 1 2 ( 1 0 1 ) 1 0 2 ( 0 1 0 1 ~ ) , ( ) , ( k k n n k k n W W k k f n n C (7) 9/4/2010 9
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
http://numericalmethods.eng.usf.edu 10 Define = = 1 0 1 ) 1 0 2 ( 0 1 0 0 1 ) , ( ) , ( k k n W k k f k n f (8) Eq. (8) can be modified to 0 2 0 0 0 0 0 1 ) , 1 ( ) , 0 ( ) , ( n W k f W k f k n f + = (9) Theoretical Development cont.
Background image of page 10
Theoretical Development cont. Hence 2 2 0 1 2 2 0 1 0 0 0 1 0 0 0 1 ) 1 , 1 ( ) 1 , 0 ( ) 1 , 1 ( ) 1 , 0 ( ) 1 , 1 ( ) 0 , 1 ( ) 0 , 0 ( ) 0 , 1 ( ) 0 , 0 ( ) 0 , 1 ( ) 1 , 1 ( ) 1 , 0 ( ) 1 , 1 ( ) 1 , 0 ( ) 1 , 0 ( ) 0 , 1 ( ) 0 , 0 ( ) 0 , 1 ( ) 0 , 0 ( ) 0 , 0 ( W f f W f W f f W f f W f W f f W f f W f W f f W f f W f W f f + = + = + = + = + = + = + = + = (10) In matrix notation, Eq.(10) can be written as = ) 1 , 1 ( ) 0 , 1 ( ) 1 , 0 ( ) 0 , 0 ( 0 1 0 0 0 1 0 1 0 0 0 1 ) 1 , 1 ( ) 0 , 1 ( ) 1 , 0 ( ) 0 , 0 ( 2 2 0 0 1 1 1 1 f f f f W W W W f f f f (11) 9/4/2010 11
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Theoretical Development cont.
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.

Page1 / 117

mws_gen_fft_spe_ppttheoreticalfourier - Numerical Methods...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online