Review: Theorem of irrelevance. Y j φ j (t) where Y j = X j + Z j for 1 j k and Y j = Z j for

Size: px

Start display at page:

Download "Review: Theorem of irrelevance. Y j φ j (t) where Y j = X j + Z j for 1 j k and Y j = Z j for"

Roy Philip Hunt
5 years ago
Views:

1 Review: Theorem of irrelevance Given the signal set { a 1,..., a M }, we transmit X(t) = j k =1 a m,jφ j (t) and receive Y (t) = j=1 Y j φ j (t) where Y j = X j + Z j for 1 j k and Y j = Z j for j>k. Assume {Z j ; j k} are iid and N (0,N 0 /2). Assume {Z j : j > k} are arbitrary rv s that are independent of {X j,z j ; j k}. Then the MAP detector depends only on Y 1,...,Y j. The error probability depends only on { a 1,..., a M }, and in fact, only on a j, a k for each j, k. Alll orthonormal expansions are the same; noise and signal outside of signal subspace can be ignored. 1

2 Next let X(t) = X n (t) where X n (t) = (n) (n) n j a m,jφ j (t) is the nth of a sequence of modulated waveforms and φ ( jn) (t) are orthonormal over j and n. If the choice of X n (t) (over signals a m ) is statistically independent from one n to another, then the optimal sequence detector is simply the optimal detector for one signal at a time. With statistical dependence between X n (t), then the error probability for optimal sequence detection is less than or equal to that for successive independent detection. This is true both for single-signal error probability and block error probability. 2

3 If {φ j (t); j Z} is an orthonormal complex set at baseband, then Ψ j1 (t) =R{2φ j (t) e 2πifct }; Ψ j2 (t) =I{2φ j (t) e 2πifct } u(t) = a j φ j (t) x(t) = a j1 Ψ j1 (t)+a j2 Ψ j2 (t) j j j y(t) = (a j1 + Z j1 )Ψ j1 (t)+(a j2 + Z j2 )Ψ j2 (t) v(t) = (a j + Z j ) a j φ j (t) j j Here {Z j ; j Z} is a sequence of iid circularly symmetric complex Gaussian rv s. Under complex linear transformations, the resulting noise rv s are Gaussian circularly symmetric. 3

4 Input Signal encoder a A Baseband Baseband modulator u(t) passband x(t) WGN + Output Baseband Passband Detector Demodulator v(t) baseband v Equivalent system y(t) Input Signal encoder a A Baseband modulator u(t) Output Detector v Baseband Demodulator v(t) + WGN; complex circularly symm. 4

5 A set of signals a 1,..., a M are orthogonal if a i, a j = Eδ ij for 1 i, j M. They span an M dimensional space and can be taken as basis vectors in R M. The mean of an orthogonal set is A =( M E,..., The set s j = a j A is a simplex code. This spans an M 1 dimensional space. The energy is E M M 1. The set ± a 1, ± a 2,...,± a M is a biorthogonal code. M E ) T 5

6 Orthogonal Simplex Biorthogonal 0,1 M =2 1, M =3 0,1,0 1,0,0 0,0,1 2/2 Note that for M 3, the lines connecting closest points are not orthogonal. 6

7 Orthogonal and simplex codes have the same error probability. The energy difference is 1 1. m Orthogonal and biorthogonal codes have the same energy but differ by about 2 in error probability. We find the ML error probability for orthogonal codes. By symmetry, doesn t depend on codeword (signal), so assume input 1. Normalize the output by W j = Y j 2/N 0. Thus the input is (α, 0,...,0) where α = 2E/N 0. Given this input, W 1 N(α, 1), W j N(0, 1) for j 2 and W 1,...,W M are independent. An error is made is W j W 1 for any 2 j M. 7

8 Pr(e) = f W1 (w 1 )Pr {W j w 1 } dw 1 j=2 If w 1 is very small, then lots of other signals look more likely; if large, then union bound is good. Let B 1,B 2,...B n be independent equiprobable events of probability p. n np for np 1 Pr( B j ) = 1 (1 p) n 1 for np > 1 j=1 n Pr( B j ) j=1 np { M { n(n 1) 2 (np) 2 p = np 2 2 np/2 for np 1 1/2 for np > 1 8

9 Pr { M j=2 (M 1)Q(w 1 ) for w 1 γ (W j w 1 1 for w 1 <γ γ Pr(e) f W1 (w 1 ) dw 1 + γ fw1 (w 1 )(M 1)Q(w 1 ) dw 1 ( ) M 1 (w 1 α) 2 = Q(α γ)+ γ 2π Q(w 1 ) exp 2 Expression on right looks Gaussian, mean α/2. 9

10 Bottom line: Choose γ = 2lnM Then Pr(e) exp ( ) (α γ) 2 2 for α/2 γ ( ) α 2 + γ 2 2 exp 4 for Let log M = b and E b = E/b. Then [ ( ) ] 2 α/2 > γ exp b E b /N 0 ln 2 for 4N b ln 2 < E b Pr(e) [ ( )] 0 exp b E 2N b ln 2 for ln 2 < E b 0 This says we can get arbitrarily small error probability so long as E b /N 0 > ln 2. This is Shannon s capacity formula for unlimited bandwidth WGN transmission. E 4N 0 N 0 10

11 Bi-Orthogonal code by Hadamard matrix Map n bit blocks to 2 n bit orthogonal sequences b = 1 b = 2 b = 3 Generate H b+1 from H b : put H b at top left, top right, lower left, and put complement H b at lower right. Each mod 2 row sum is a row - half ones. Follow by antipodal modulation. 11

12 Convolutional Encoding U j,1 Input bits D j Dj 1 Dj 2 R = 1/2,n =2 U j,2 U j,1 = D j D j 1 D j 2 U j,2 = D j D j 2 It needs n bits at end of block to return to state 0. Viterbi algorithm used for decoding; complexity 2 n. 12

13 Input bits D j D j 1 Dj 2 U j,1 R = 1/2,n =2 U j,2 00 State

14 State Viterbi decoding: At each epoch, decode conditional on each possible assumed state. Maintain only the survivor at each state; each decoding step is a binary decision. 14

15 WIRELESS COMMUNICATION Wireless: radiation between antennas. Much more difficult than wires. Permits motion and temporary locations. Avoids mazes of wires NEW PROBLEMS: 1. Channel changes with time 2. Interference between channels 15

16 Started by Marconi in 1897; Many false starts We will concentrate on Cellular Networks This includes most features of other systems. Many mobiles, Few base stations. Mobile Base station MTSO Wired network Whatever 16

17 Hexagon Cells Real Cells Base Stations MTSO 17

18 Cellular Network is Appendage to Wire Network Major Problems: Outgoing: Find Best Base station Ingoing: Find Mobile Multiple mobiles send to same base station. This is called the reverse channel or a multiaccess channel Base station sends to multiple mobiles. this is called the forward channel or a broadcast channel. 18

19 Wireless Systems are now digital (Binary In terface) Source either analog or digital. Cellular systems developed for voice But major issues quite different for voice and data 19

20 OTHER WIRELESS SYSTEMS: Broadcast Systems Wireless LANs (often in home or office) Adhoc Networks Standardization is a major problem for all wireless systems Particularly a problem for cellular because of roaming. Will voice and data wireless networks merge into one, or will they evolve into separate networks? 20

21 Is there a large market for high speed mobile data? We study more technical issues in what follows.

22 PHYSICAL MODELING Wireless uses bandwidths of KH to a few MH in bands of a few GH. Cellular ranges are small, a few KM or less Narrow band; WGN assumption good, but new problems are fading and interference. EM equations are too difficult to solve and constantly changing. Very different modeling questions arise in the placement of base stations from those in the design of mobiles and base stations. Look at idealized models for clues 21

23 Consider fixed antenna in free space: Response at x = (r, θ, ψ) to sinusoid at f: [ ] 1 r E(f, t, x) = R α s (x, f)) exp{2πif (t )} r c Note 1/r attenuation; think spheres Receiving antenna alters field; doesn t depend on (r, θ, ψ). Define H(f) = α(θ, ψ, f ) exp{ 2πifr/c} r E r (f, t, u) = R [H(f) exp{2πift}] Linearity holds but not time invariance. 22

24 MIT OpenCourseWare Principles of Digital Communication I Fall 2009 For information about citing these materials or our Terms of Use, visit:

Wireless Systems are now digital (Binary Interface)

Wireless Systems are now digital (Binary Interface) Source either analog or digital. Cellular systems developed for voice: need small fixed delay; fixed data rate; not high reliability. Data: delay less