Digital Communications I: Modulation and Coding Course. Term Catharina Logothetis Lecture 8

Transcription

1 Digital Communication I: odulation and Coding Coure Term 3-8 Catharina Logotheti Lecture 8

2 Lat time we talked about: Some bandpa modulation cheme -A, -SK, -FSK, -QA How to perform coherent and noncoherent detection Lecture 8

3 xample of two dim. modulation 6QA ψ t SK ψ t QSK 5 ψ t ψ t ψ t ψ t Lecture

4 Today, we are going to talk about: How to calculate the average probability of ymbol error for different modulation cheme that we tudied? How to compare different modulation cheme baed on their error performance? Lecture 8 4

5 rror probability of bandpa modulation Before evaluating the error probability, it i important to remember that: The type of modulation and detection coherent or noncoherent determine the tructure of the deciion circuit and hence the deciion variable, denoted by. The deciion variable,, i compared with - threhold, correponding to deciion region for detection purpoe. ψ t rt ψ t N T T r r N r r N r r Deciion Circuit Compare with threhold. mˆ Lecture 8 5

6 rror probability The matched filter output obervation vector r i the detector input and the deciion variable i a f r function of r, i.e. For A, QA and FSK with coherent detection r For SK with coherent detection r For non-coherent detection -FSK and DSK, r We know that for calculating the average probability of ymbol error, we need to determine r r lie inide Z i i ent r atifie condition C Hence, we need to know the tatitic of, which depend on the modulation cheme and the detection type. i i ent Lecture 8 6

7 rror probability AWGN channel model: r i + The ignal vector i ai, ai,..., ain i determinitic. The element of the noie vector n n, n,..., nn are i.i.d Gauian random variable with ero-mean and variance N /. The noie vector' pdf i n pn n exp N / π N N The element of the oberved vector r r, r,..., r N are independent Gauian random variable. It pdf i r i pr r i exp N / π N N n Lecture 8 7

8 rror probability BSK and BFSK with coherent detection: BSK B ψ t b b b Q ψ t / N / BFSK ψ t b b b ψ t B Q b N B Q b N Lecture 8 8

9 rror probability Non-coherent detection of BFSK / T co ω t T / T in ω t r r r + Deciion variable: Difference of envelope rt / T co ω t T T r r + - Deciion rule: if T >, mˆ if T <, mˆ mˆ / T in ω t r r + T r Lecture 8 9

10 Lecture 8 rror probability cont d Non-coherent detection of BFSK Similarly, non-coherent detection of DBSK [ ] > > > > + >, r, r r r r d p d p d p B exp N b B exp N b B Rayleigh pdf Rician pdf

11 Coherent detection of -A Deciion variable: rror probability. r 4-A 3 g g g g ψ t ψ t rt T r L detector Compare with - threhold mˆ Lecture 8

12 Lecture 8 rror probability. Coherent detection of -A. rror happen if the noie,, exceed in amplitude one-half of the ditance between adjacent ymbol. For ymbol on the border, error can happen only in one direction. Hence: g e g e g m m e r n r n m r n < > < < > r and r ; for r 6 log N Q b g b 3 log m r n Gauian pdf with ero mean and variance / N > < + > + > r r r r N Q dn n p n n n n g n g g g g m m e g

13 rror probability Coherent detection of -QA ψ t ψ t r 6-QA L detector 9 ψ t rt ψ t T r Compare with threhold L detector arallel-to-erial converter mˆ T Compare with threhold Lecture 8 3

14 rror probability Coherent detection of -QA -QA can be viewed a the combination of two modulation on I and Q branche, repectively. No error occur if no error i detected on either the I or the Q branch. Conidering the ymmetry of the ignal pace and the orthogonality of the I and Q branche: A C rno error detected on I and Q branche rno error detected on I and Q branche 4 Q 3log N b rno error on Irno error on Q rno error on I Lecture 8 4 Average probability of ymbol error for A

15 rror probability Coherent detection of SK ψ t SK 5 ψ t ψ t r rt ψ t T r arctan r φˆ Compute φ φˆ i Chooe mallet mˆ T r Deciion variable φˆ r Lecture 8 5

16 Lecture 8 6 rror probability Coherent detection of SK The detector compare the phae of obervation vector to - threhold. Due to the circular ymmetry of the ignal pace, we have: where It can be hown that φ φ π π φ d p c m m c C / / ˆ N Q π in N Q b π in log or ; in exp co ˆ π φ φ φ π φ φ N N p

17 rror probability Coherent detection of -FSK ψ t rt ψ t T T r r r r r r L detector: Chooe the larget element in the oberved vector mˆ Lecture 8 7

18 rror probability Coherent detection of -FSK The dimenion of the ignal pace i. An upper bound for the average ymbol error probability can be obtained by uing the union bound. Hence: or, equivalently Q N Q log N b Lecture 8 8

19 Bit error probability veru ymbol error probability Number of bit per ymbol For orthogonal -ary ignaling -FSK lim k B k B k / For -SK, -A and -QA k log B k for < < Lecture 8 9

20 robability of ymbol error for binary modulation Note! The ame average ymbol energy for different ie of ignal pace b / N db Lecture 8

21 robability of ymbol error for -SK Note! The ame average ymbol energy for different ie of ignal pace b / N db Lecture 8

22 robability of ymbol error for -FSK Note! The ame average ymbol energy for different ie of ignal pace b / N db Lecture 8

23 robability of ymbol error for -A Note! The ame average ymbol energy for different ie of ignal pace b / N db Lecture 8 3

24 robability of ymbol error for - QA Note! The ame average ymbol energy for different ie of ignal pace b / N db Lecture 8 4

25 xample of ample of matched filter output for ome bandpa modulation cheme Lecture 8 5