What is a Communications System?

Transcription

1 Wha is a ommuiaios Sysem? Aual Real Life Messae Real Life Messae Replia Ipu Sial Oupu Sial Ipu rasduer Oupu rasduer Eleroi Sial rasmier rasmied Sial hael Reeived Sial Reeiver Eleroi Sial Noise ad Disorio Modulaio/odi Sial o Noise Raio SNR Demodulaio Reduday apaiy Error orreio

2 lassifiaio of Sials oiuous-ime ad Disree-ime Sials Aalo ad Diial Sials Deermiisi ad Radom Sials Periodi ad Aperiodi Sials Eery ad Power Sials

3 oiuous vs. Disree ime Aalo vs. Diial ime Value oiuous oiuous oiuous Disree 3 Disree oiuous Aalo Diial 5 Disree Disree

4 Some Eamples

5 Deermiisi vs. Radom Sials

6 Effe of SNR

7 Periodi vs. Aperiodi Sials Sial is Periodi if : Period of he Periodi Sial = Frequey of he Periodi Sial = f = / If he sial is o Periodi, i will be Aperiodi

8 Some Eamples 3si4 si si6

9 Size of he Sial? Eery of Sial E lim / / Power of Sial ime Averae Eery P RMS value of Sial P

10 Eample of Eery ad Power Sials Neiher e / Eery Sial u e / Power Sial os Power Sial os os

11 Eample eiher eery or power sial e / E e / e e

12 Eample eery sial u e / E / e e e

13 os Eample 3 power sial / / lim P / / os lim / / ] os [ lim / / / / ] os lim lim

14 lim / / lim / / os ] lim lim P For a siosoidal sial reardless of frequey ad phase shif RMS value /

15 Eample 4 power sial os os / / lim P / / / / / / os os lim os lim os lim / / os os lim

16 / / os os lim b a b a b a os os os os Hi -> / / / / ] os[ lim ] os[ lim P P Power of he sum of he wo siosoid sials wih disi frequeies is equal o he sum of he powers of he idividual sials

17 Geeralizi he Resul os P I is alled Parseval s heorem

18 Some Impora Operaios o Sials ime Shifi a ime Sali a ime Iversio

19 Ui Impulse Fuio Defiiio: Sampli Propery or a a

20 Ui Sep Fuio Defiiio: u a you represe u i erms of ui impulse fuio? u d

21 ompoes of a Sial Le s approimae a sial wih aoher sial Error of approimaio is: e Eery size of error is E e e [ ] For bes approimaio, eery eeds o be miimized

22 o miimize eery, a eessary odiio is d de e ] [ d d d d d d d d or E is he projeio of o

23 Eample - si? We kow E E si si 4 si si si 4

24 Eample -? - E We kow E has a full projeio o boh are same

25 Eample 3? - E We kow E has a full eaive projeio o or ad are opposie o eah oher

26 Eample 4 -? E We kow E? does o have ay projeio o or ad are orhooal o eah oher

27 orrelaio bewee Sials We already kow, if E Le s defie a orrelaio o-effiie bewee wo fuios, ad, as Or o eeralize E E E E Opposie Orhooal Iself

28 Eery of Sum of Orhooal Sials z z E E E If ad are orhooal, he By defiiio: E z ] [ E E

29 Orhooal Sial Se m m E m Le s defie a sial se,,., N, suh ha he,,,., N are alled orhooal sial se ad if E is =, he he se is alled orhoormal Now le s assume N N N E he for bes approimaio i.e., miimized error eery

30 If he orhooal se is omplee, he error eery ->, i.e., approimaio haes o equaliy N N N he above equaio is alled eeralized Fourier Series Wha abou eery of? Parseal s heorem N N N E E E E E E

31 A Eample of omplee Orhooal Sial Se {, os,os,os3... si,si,si 3,...} f Follows by: os os m / m m si si m / m m si os m for all m ad

32 Le s assume:... si 3 si si... os3 os os 3 3 b b b a a a a ] si os [ b a a E,,3... os a,,3... si b a Remember: herefore, rioomeri Fourier Series

33 oep of represei a periodi sial wih a summaio of siusoids

34 ompa rioomeri Fourier Series a [ a os b si ] os ompa rioomeri Fourier Series Where a a b b a Also alled Fourier Sperum of periodi sial a

35 Eample se f Hz Wihou ay mahemaial alulaios, a b si

36 Eample o. se f Hz a os a.5 os os os 4 os.5.5

37 .5 4 os 4 os os 4 os.5.5 4si.5 4si.5.5 a 4 si si si 4 si

38 Eample o. se f Hz a 4 si a 4 a a3 4 3 a 4 a a 6

39 Eample o. se f Hz a [ a os b si ] os os3 os5 os os 4 os3 3 4 os5 5 4 os7... 7

40 y Eample se f Hz y We kow ha os os3 os5 os herefore, y os os3 os5 os

41 y Eample 3 f We kow ha wih ime period = se y os os3 os5 os herefore, wih ime period of y os os3 os

42 Eample f a os Remember defiiio of impulse fuio

43 Eample o. f a os Remember eeded sampli properies of impulse fuio b si

44 a b a a a b a a b a Eample o. f

45 Eample o. f os os

46 Eample o. ime Domai Sial f Fourier Sperum 3 4

47 Eery of Sial is defied Eery of Sial Revisied E o aommodae omple sials ad o E Beause for real sials * for omple sials Similarly, * beomes omple sials

48 Aoher Eample of orhooal Sial Se aai periodi e j,,,... f e j e * j m e jm os m j si m os m j si m m m

49 Epoeial Fourier Series D e j By orhooaliy or D D e e j j *

50 os j j e e j j j j e e e e j j e D D e D j D e D j e D ompa o Epoeial Fourier Series

51 Le s look bak o he Eample of Impulse rai f os os e D j e D j D

52 ime Domai Sial f Fourier Sperum 3 4 Epoeial Fourier Sperum D