Orthogonal Signals With orthogonal signals, we select only one of the orthogonal basis functions for transmission:

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1 4..4 Orthogonal, Biorthogonal and Simplex Signal 4.- In PAM, QAM and PSK, we had only one ai function. For orthogonal, iorthogonal and implex ignal, however, we ue more than one orthogonal ai function, o N-dimenional Example: the Fourier ai; time-tranlated pule; the Walh-Hadamard ai. o We ve ecome ued to SER getting wore quickly a we add it to the ymol ut with thee orthogonal ignal it actually get etter. o The drawack i andwidth occupancy; the numer of dimenion in andwidth W and ymol time T i N WT o So we ue thee et when the power udget i tight, ut there plenty of andwidth. Orthogonal Signal With orthogonal ignal, we elect only one of the orthogonal ai function for tranmiion: The numer of ignal M equal the numer of dimenion N.

2 Example of orthogonal ignal are frequency-hift keying (FSK), pule poition modulation (PPM), and choice of Walh-Hadamard function (note that with Fourier ai, it FSK, not OFDM). 4.- Energy and ditance: o The ignal are equiditant, a can e een from the ketch or from i E j = E location i location j o i j E M E ke dm = = log ( ) = = in, i, j

3 Effect of adding it: 4.-3 o For a fixed energy per it, adding more it increae the minimum ditance trong contrat to PAM, QAM, PSK. o But adding each it doule the numer of ignal M, which equal the numer of dimenion N and that doule the andwidth! Error analyi for orthogonal ignal. Equal energy, equiproale ignal, o receiver i o Error proaility i ame for all ignal. If wa ent, then the correlation vector i c E + n n = n M with real component.

4 o Aume wa ent. The proaility of a correct ymol deciion, conditioned on the value of the received, i c ( ) ( ) ( ) = < < < Pc ( c) P n c n3 c nm c M c Q N = o Then the unconditional proaility of correct ymol detection i c = M ( ) Pc Q pc c dc N0 M c ( c ) N 0 N0 N0 = Q exp E dc π M ( ( )) exp ( ) = Qu u γ du π Need a numerical evaluation. o The unconditional proaility of ymol error (SER) i P e = P c

5 Now for the it error rate. o All error equally likely, at 4.-5 Pe Pe =. No point in Gray coding. k M k o Can get i it error in equiproale way, o i k k P k P k Pe = k i k k i= i = i= i k k = k P k P Aout half the it are in error in the average ymol error. The SER for orthogonal ignal i triking: pro of ymol error Symol Error Pro, Orthogonal Signal M = M = 4 M = 8 M = The SER improve a we add it! But the andwidth doule with each additional it. SNR per it, gamma_ (db)

6 Why doe SER drop a M increae? 4.-6 o All point are eparated y the ame d = log ( M E ). Since d increae linearly with k = log ( M), the pairwie error proaility (etween two pecific point) decreae exponentially with k, the numer of it per ymol. o Offetting thi i the numer M- of neighour of any point: M- way of getting it wrong, and M increae exponentially with k. o Which effect win? Can t get much analytical traction from the integral expreion two page ack. So fall ack to a union ound. Union ound analyi: o Without lo of generality, aume wa tranmitted. o Define Em, m=,, M a the event that r i cloer to m than to.

7 o Note that E m 4.-7 i a pairwie error. It doe not imply that the receiver deciion i m. In fact, we have event E,, EM and they are not mutually excluive, ince r could lie cloer to two or more point than to : o The overall error event E = E E3 EM, o the SER i ounded y the um of the pairwie event proailitie [ ] [ ] [ ] P = P E = P E E E P E e 3 M m m= d = ( M ) Q = ( M ) Q log ( M) γ N0 γ k ln() kγ k e < e M ( ) A general expanion i [ ] [ ] M M M + P Ai Aj Ak i= j= k= j i k j k i M M M P A A A = P A P A Aj M i i i= i= i j= j i etc.

8 o From the ound, if ( ) 4.-8 γ > ln =.386, then loading more it onto a ymol caue the upper ound on SER to drop exponentially to zero. o Converely, if γ < ln( ), increaing k caue the upper ound on SER to rie exponentially (SER itelf aturate at ). o Too ad the dimenionality, the numer of correlator and the andwidth alo increae exponentially with k. Thi cheme i called lock orthogonal coding. Threhold are characteritic of coded ytem. Rememer that thi analyi i aed on ound 0 SER and Bound, Orthogonal Signal True SER are olid line, ound are dahed. pro of ymol error M = M = M = 4 M = 4 M = 8 M = 8 M = 6 M = SNR per it, gamma_ (db) Upper ound i ueful to how decreaing SER, ut i too looe for a good approximation.

9 4.-9 Biorthogonal Signal If you have coherent detection, why wate the other ide of the axi? Doule the numer of ignal point (add one it) with ± E. Now M = N, with no increae in andwidth. Or keep the ame M and cut the andwidth in half. Energy and ditance: All ignal point are equiditant from i = E = d (the ame a orthogonal ignal) i k min except one the reflection through the origin which i farther away i i = E

10 Symol error rate: The proaility of error i mey, ut the union ound i eay. A ignal i equiditant from all other ignal ut it own complement. o So the union ound on SER i ( ) ( ) ( ) ( ) P ( M ) Q γ + Q γ ( M ) Q γ econd term redundant (Sec'n 4.5.4) = ( M ) Q log ( M) γ which i lightly le than orthogonal ignaling, for the ame M and ame γ. The major enefit i that iorthogonal need only half the andwidth of orthogonal, ince it ha half the numer of dimenion. o And the it error proaility i log ( M ) P P

11 Simplex Signal 4.-3 The orthogonal ignal can e een a a mean value, hared y all, and ignal-dependent increment: The mean ignal (the centroid) i M m =. M m = o The mean doe not contriute to ditinguihaility of ignal o why not utract it? = m m M M m = E M location m M where E i the original ignal energy.

12 4.-3 o Removing the mean lower the dimenionality y one. After rotation into tidier coordinate: They till have equal energy (though no longer orthogonal). That energy i E = m = E M + = E M M M The energy aving i mall for larger M.

13 The correlation among ignal i: ( m, ) n Re[ ρ mn ] = = M = m n M M A uniform negative correlation. for m n The SER i eay. Tranlation doen t change the error rate, o the SER i that of orthogonal ignal with a M ( M ) SNR oot Vertice of a Hypercue and Generalization More ignal defined on a multidimenional pace. Unlike orthogonal, we will now allow more than one dimenion to e ued in a ymol. Vertice of a hypercue i traightforward: two-level PAM (inary antipodal) on each of the N dimenion: o With the Fourier ai, it i BPSK on each frequency at once a imple OFDM o With the time tranlate ai, it i a claical NRZ tranmiion:

14 o Signal vector: m m = m mn with mn =± E N for m=, M and M = N. o In pace, it look like thi o Every point ha the ame ditance from the origin, hence the ame energy m log ( ) = E = M E = N E

15 o Minimum ditance occur for difference in a ingle coordinate: m E = N n E = N E dmin = m n = = E N More generally, for d H diagreement (Hamming ditance), H m n = d E = dhe N o What are the effect on d min and andwidth if we increae the numer of it, keeping E fixed? It eay to generalize from inary to PAM, PSK, QAM, etc. on each of the dimenion: o With the Fourier ai, it i OFDM o With time tranlate, it i the uual erial tranmiion.

16 Error analyi: All point have ame energy E = N E Euclidean ditance for Hamming ditance h i d = he o If laeling i done independently on different dimenion (ee ketch), then the independence of the noie caue it to decompoe to N independent detector. So the following tructure P NQ ( γ ) ( ) = log ( MQ ) γ P depend on lael ecome P i meaningle P ( ) = Q γ, jut inary antipodal

17 4.-37 Generalization: uer multilevel ignal (PAM) in each dimenion. Again, if laeling i independent y dimenion, then it jut independent and parallel ue, like QAM. Not too exciting. Yet. Thee ignal form a finite lattice. Generalization: ue a uet of the cue vertice, with point elected for greater minimum ditance. Thi i a inary lock code. o Advantage: greater Euclidean ditance o Diadvantage: lower data rate