Conceptual similarity to linear algebra

Transcription

1 Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3-D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ), they are orthogonal i.e., perpendicular to each other if, and only if, x y = x 1 y 1 + x 2 y 2 + x 3 y 3 =0 called dot product (or inner product) 3-D: (1, 0, 0), (0, 1, 0), (0, 0, 1) are orthogonal also basis of 3-D called orthonormal if dot product with itself is 1

2 Lots of other orthogonal basis vectors For example: (5, 2, 0), (2, 5, 0), (0, 0, 1) are mutually orthogonal but not orthonormal how to make them orthonormal? Relevance to networking: In CDMA (code division multiple access) for example, used by Sprint and Verizon for wireless cellular in the U.S. (5, 2, 0), (2, 5, 0), (0, 0, 1) are called codes one code per user 3-D codes: 3 users (say Bob, Mira, Steve)

3 Suppose each user wants to send a single bit Bob: 1, Mira: 0, Steve: 0 Bob s cell phone: send 1 (5, 2, 0) to base station (cell tower) Mira s cell phone: send 1 (2, 5, 0) to base station Steve s cell phone: send 1 (0, 0, 1) common convention: 1 for bit 1, 1 forbit0 Base station receives: (3, 7, 1) ( 1 (5, 2, 0) ) + ( 1 (2, 5, 0) ) + ( 1 (0, 0, 1) ) How can base station find what bit Bob has sent?

4 Base station: compute the dot product of what it has received, (3, 7, 1), and the code of Bob, (5, 2, 0) (5, 2, 0) (3, 7, 1) = = 29 positive: hence bit 1 what s special about 29? To find out what Mira has sent: (2, 5, 0) (3, 7, 1) = = 29 negative: hence bit 0 To find out what Steve has sent: (0, 0, 1) (3, 7, 1) = = 1 negative: hence bit 0 why does this work?

5 Base station decoding Bob s bit: (5, 2, 0) (3, 7, 1) Since (3, 7, 1) = ( 1 (5, 2, 0) ) + ( 1 (2, 5, 0) ) + ( 1 (0, 0, 1) ) (5, 2, 0) (3, 7, 1) equals ( 1 (5, 2, 0) (5, 2, 0) ) + ( 1 (5, 2, 0) (2, 5, 0) ) which equals ( 1 (5, 2, 0) (5, 2, 0) ) + ( 1 (5, 2, 0) (0, 0, 1) ) the two interference terms are nullified orthogonality! Same holds when computing Mira s bit and Steve s bit CDMA has additional twists (discussed in wireless) but the above is essential idea

6 Back to orthogonal FDM (OFDM) key idea: use carrier waves that are orthogonal Dot product of two vectors x =(x 1,...,x n )andy = (y 1,...,y n ) n x y = x i y i i=1 Dot product of two sinusoids x(t) = sin f x t and y(t) = sin f y t x(t) y(t) = again: just a sum of products (sin f x t) (sin f y t) dt More generally: x(t) y(t) = eif xt e if yt dt since Fourier transform involves complex sinusoids

7 User 1 uses carrier wave y 1 (t) to transmit bit stream (high and low) given by x 1 (t) z (t) 1 y (t) 1 x (t) 1 Same for users 2 and 3 Suppose carrier waves y 1 (t), y 2 (t), y 3 (t) areorthogonal

8 Then receiver sees z 1 (t)+z 2 (t)+z 3 (t) whichis 3 x k (t)y k (t) k=1 To decode what user 1 has sent, receiver computes dot product with y 1 (t) y 1 (t) ( 3 k=1 ) x k (t)y k (t) 3 = x k (t) ( y 1 (t) y k (t) ) k=1 = x 1 (t) ( y 1 (t) y 1 (t) ) = x 1 (t) last steps holds if also orthonormal But look at Fourier transform formula (lecture notes, part 2): just taking dot product!

9 Root of FDM solution: Joseph Fourier 18th century idea ( old technology ) Fourier analysis engineering bread and butter worth knowing for its own sake (great idea)

10 Fourier s key insight: A complicated looking signal s(t) whose shape (i.e., strength) varies over time is just the sum of very simple building blocks sinusoids

11 Thus: may require adding many sinusoids of di erent frequencies key caveat: before adding sinusoid with frequency f, multiply its magnitude by a weight therefore complicated looking s(t) is just the weighted sum of sinusoids

12 Some conceptual similarity to periodic table and matter elements of periodic table: building blocks (sinusoids) matter: complicated looking signal H 2 O: 2 parts H and 1 part O C 8 H 10 N 4 O 2 of course, matter has additional structure: not simple weighted sum

13 Obvious consequences: The value of the weight f of sinusoid f indicates how important sinusoid f is For example, if f =0thensinusoidoffrequencyf is not needed at all for creating s(t) Since there are an infinite number of frequencies (from 0 to ) the weighted sum may entail an infinite number of sinusoids not relevant for FDM: why?

14 Fourier s key insight is accompanied by a key technical contribution: If given some complicated looking signal s(t), then for any sinusoid f Fourier provides a simple formula for finding its weight f a way to decompose into building blocks

15 Let s make use of Fourier s insight for enabling broadband communication: point-to-point link from A to B A wishes to send 3 bits to B in parallel using three carrier frequencies f 1, f 2,andf 3 say 3 bits: 1, 0, 1 carrier frequencies: 1 Hz, 2 Hz, 3 Hz how to do it?

16 Fourier s conceptual and technical contribution in more precise language (aka math): 1. Complicated looking signal s(t) is weighted sum of sinusoids s(t) = 1 2 f e ift df called Fourier expansion integral is just continuous sum recall: e ift = cos ft+ i sin ft Euler s formula why complex sinusoids involving i = 1?

17 2. Given s(t) andfrequencyf, howtofindweight f : f = s(t)e ift dt another weighted sum called Fourier transform algorithm to compute Fourier transform quickly: fast Fourier transform (FFT) see algorithms book