Demodulation of Digitally Modulated Signals

Transcription

1 Addiional maerial for TSKS1 Digial Communicaion and TSKS2 Telecommunicaion Demodulaion of Digially Modulaed Signals Mikael Olofsson Insiuionen för sysemeknik Linköpings universie, Linköping November 29

2 Noe: This documen is supposed o be addiional maerial for he courses TSKS1 Digial Communicaion and TSKS2 Telecommunicaion, given a Linköping Universiy. The main references in hose courses are he compendiums Inroducion o Digial Communicaion [1] and Telecommunicaion Mehods [2], wrien by Mikael Olofsson, Thomas Ericson, Rober Forchheimer and Ulf Henriksson (only he firs one). We follow he noaion in hose compendiums, and relaions from here are used here wihou proof. Demodulaion of Digially Modulaed Signals c 29 Mikael Olofsson Insiuionen för sysemeknik Linköpings universie Linköping This documen is wrien using L A TEX2ε. xfig.org). The figures are creaed using Xfig (från

3 Conens 1 Inroducion 1 2 Demodulaion using a Correlaion Receiver 1 3 Demodulaion using Mached Filers 1 4 Eye Paerns 3

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5 1. Inroducion 1 1 Inroducion Digial modulaion is reaed in he course compendiums. In [1], he reamen is fairly deailed, while in [2], he reamen is fairly brief. Some hings ha are of ineres are missing here. This maerial is supposed o fill ha lack, by reaing a number of aspecs of demodulaion of digially modulaed signals. Secions 2 and 3 are primarily aimed a hose reading [2]. 2 Demodulaion using a Correlaion Receiver Le x() be a signal ha is zero ouside [,T]. We express x() in he wo basis funcions φ () and φ 1 () as x() = x φ () + x 1 φ 1 (), i.e. is vecor represenaion wih respec o hose basis funcions are ( ) x x = On page 51 in [1] and page 18 in [2], i is noed ha he relaions x = x 1 x()φ () d and x 1 = x()φ 1 () d hold. The ask of he receiver is o exrac he coefficiens x and x 1 from x(). Tha can be done by performing exacly he calculaions above. Such a receiver is usually called a correlaion receiver, and you can find i ogeher wih he modulaor and channel in Figure 1. Noe ha ha figure allows for more han wo basis funcions, which is perfecly possible even hough i is no menioned in [2]. 3 Demodulaion using Mached Filers An LTI filer wih impulse response h j (), defined as h j () φ j (T ) (1) is said o be mached o he basis funcion φ j (). Graphically, we can see h j () as φ j () refleced in = T/2. Thus, h j () is also zero ouside [,T]. Le x() be he inpu o his filer, and le y j () be is oupu. Then we have y j () = (x h j )(),

6 2 Demodulering av digial modulerade signaler φ 1 () φ 1 () correlaor s i1 X 1 W() s i () X() φ N () φ N () correlaor s in X N Modulaor Channel Correlaion receiver Figure 1: Modulaor, channel, and correlaion receiver. φ 1 () s i1 W() φ 1 (T ) = T X 1 s i () X() φ N () s in φ N (T ) = T X N Modulaor Channel Mached filer receiver Figure 2: Modulaor, channel, and mached filer receiver.

7 4. Eye Paerns 3 where denoes convoluion as usual. This gives us y j () = x(τ)h j ( τ)dτ = T x(τ)φ j (T + τ)dτ, where we have idenified h j () using Equaion 1, and where he finie inegraion limis are based on he assumpion ha φ j () is zero ouside [,T]. Consider he ime insance = T. Then we have y j (T) = x(τ)φ j (τ)dτ = x j, i.e. exacly he j-h coefficien in he waned vecor. Using words, we can exrac he coefficien x j from x() by filering x() using a filer ha is mached o φ j (), and hen sample he oupu in he ime insance = T. A receiver based on his approach is given in Figure 2. This figure also allows for more han wo basis funcions. 4 Eye Paerns The principles of recepion ha we have described so far demand ha he sender and he receiver are compleely syncronous, and he error probabiliy expressions given in [1] and [2] for differen signal consellaions demand his complee syncronism. A useful ool o deermine how he error probabiliies depend on non-syncronism are eye paerns. We use hese eye paerns o sudy he oupu of he mached filers in Figure 2. An eye paern can mos easily be described by how you measure i. You synchronize a memory oscilloscope so ha i sars measuring in he beginning of a signal inerval. Then you produce a random sequence of symbols ha you modulae using he signal consellaion ha is a hand and demodulae according o Figure 2, and measure he oupu from he mached filer ha you are ineresed in. Wha you hen see are all possible varians of oupus. As an example, consider anipodal signalling, where we have he wo signals s () = A, < T, s 1 () = A, < T. We have a one-dimensional signal consellaion using he basis funcion φ() = 1/ T, < T, and a filer ha is mached o his basis funcion has impulse response h() = φ(t ) = 1/ T, < T.

8 4 Demodulering av digial modulerade signaler s () s 1 () T T y () y 1 () 2T Figure 3: The signals s () and s 1 () and he corresponding oupus y () and y 1 () from he mached filer h(). s() s () s ( T) T 3T s 1 ( 2T) s 1 ( 3T) s ( 4T) 5T y() y () y ( T) y ( 4T) T 2T 3T 4T 5T 6T y 1 ( 2T) y 1 ( 3T) Figure 4: A possible inpu and he corresponding oupu from he mached filer. The ineresing par of he oupu is drawn wih a solid line. T 2T 3T Figure 5: The eye paern, i.e. all possible oupu ransiions during an inerval of duraion T, repeaed a number of inervals.

9 References 5 The oupus from his filer ha correspond o he wo inpus are hen y () = (s h)(), y 1 () = (s 1 h)(). In Figure 3, he signals s (), s 1 (), y () and y 1 () are displayed. When we are communicaing using his signal consellaion, we do no send jus one signal, bu a whole sequence of such signals corresponding o a sequence of bis. This can be described as a sum of shifed signals, chosen among s () and s 1 (). Similarily, since he filer is an LTI sysem, he oupu will be he corresponding sum of shifed oupus. In figur 4, he possible inpu s() = s () + s ( T) + s 1 ( 2T) + s 1 ( 3T) + s ( 4T), and he corresponding oupu y() = y () + y ( T) + y 1 ( 2T) + y 1 ( 3T) + y ( 4T) are displayed, or a leas he ineresing par for T < < 5T. Finally, all ineresing ransiions in he middle of a communicaion signal in an inerval of duraion T are displayed in Figure 5. If we sample a he correc ime insances (i.e. nt), hen we ge maximal separaion of he wo possible symbols. If we sample a a slighly wrong ime insance, hen some samples will be closer o he decision hreshold, which is zero. Tha means ha he effecive disance beween he signal poins is decreased, and he resuling error probabiliy is herefore increased. On op of ha, he error probabiliy will depend on previously sen signals. The way he lines approach he ideal sample insances nt deermine how sensiive he demodulaion is o syncronizaion errors, and ha in urn depends on he acual signals. From his poin of view, i is no only imporan o choose signal poins, bu also o choose suiable basis funcions. References [1] Mikael Olofsson, Thomas Ericson, Rober Forchheimer & Ulf Henriksson, Inroducion o Digial Communicaion, Deparmen of Elecrical Engineering, Linköping Universiy, Augus 28. [2] Mikael Olofsson, Thomas Ericson & Rober Forchheimer, Telecommunicaion Mehods, Deparmen of Elecrical Engineering, Linköping Universiy, January 27.