Review Topics from Chapter 3&4. Fourier Series Fourier Transform Linear Time Invariant (LTI) Systems Energy-Type Signals Power-Type Signals

Transcription

1 Rviw opics from Chapr 3&4 Fourir Sris Fourir rasform Liar im Ivaria (LI) Sysms Ergy-yp Sigals Powr-yp Sigals

2 Fourir Sris Rprsaio for Priodic Sigals Dfiiio: L h sigal () b a priodic sigal wih priod. () ca b padd i rms of h compl poial sigals j j h cofficis ar calld h Fourir sris cofficis of h sigal (). hs ar, i gral, compl umbrs. [rad] is calld h fudamal frqucy. h frqucis of h compl poial sigals ar mulipls of his frqucy.

3 Alraiv Rprsaios for Ral Priodic Sigals si j cos j A jb A B A B arcgb A whr cos Prov i! Mai sps i h proof : j j j j R 3

4 Alraiv Rprsaios for Ral Priodic Sigals (Co d) si j cos j A B A cos B si Prov i! Mai sps i h proof : R A jb cos j j R si If () is a v fucio, i.. ()=(-), B =. If () is a odd fucio, i.. ()=-(-), A =. Prov hm! 4

5 Eampl: Priodic Ga Fucio () A df, rc, / /.. - -τ/ τ/ () is a v priodic fucio. ()=(-) A / cos cos For =, h abov givs / A si A / / A Alraivly, sarig from gral prssio ad usig L Hospial rul: si A A si A A lim lim lim cos Diffriaio w.r.. 5

6 sa df si sa si ovr argum fucio A si A sa Assum 4 - Summary: h spcrum of priodic sigals cosiss of a sris of impuls fucios, which is a discr spcrum

7 Spcrum for various valus of, --- fid /. /. /.5 Obsrvaios: h ampliud is proporioal o /. h spacig bw lis is proporioal o /. Zro crossigs of h spcrum rmai h sam. (o dp o ). 7

8 Spcrum for various valus of, --- fid.. - τ Obsrvaios: h ampliud is proporioal o τ... - τ.. - τ h spcrum sprads as τdcrass. hr is a ivrs rlaioship bw puls wih i im (τ) ad frqucy sprad of h spcrum. 8

9 Fourir rasform h Fourir sris is a mas for padig a priodic sigal i rms of compl poials. Now, cosidr a apriodic fucio. How ca w prss i as a sum of poial sigals? Cosruc a w priodic fucio () basd o h origial apriodic sigal (). () (). - -/ /. lim 9

10 () is a priodic sigal ad ca b rprsd by Fourir sris. j j whr j j df df Dfiig df lim lim As bcoms larg, bcoms smallr. I h limiig cas, h discr lis i h spcrum mrg ad frqucy spcrum bcoms coiuous. j j d j Rima Igral

11 Fourir rasform F j Ivrs Fourir rasform F - j d is also kow as spcral-dsiy fucio of (). F Fourir rasform Pair Rlaioship bw Fourir Sris & Fourir rasform

12 Eampl: Ga fucio df, / rc, / () is a v apriodic fucio. A rc A -τ/ τ/ / cos A cos si / A / / A si A sa Rlaio o Fourir Sris: W had obaid Fourir sris for a priodic ga fucio (s prvious ampl). A sa A sa

13 A sa A Obsrvaio: Compar h spcrum o priodic sigals. Apriodic sigals hav coiuous spcrum. 3

14 Fourir rasform for Priodic Sigals A priodic sigal ca b rprsd by is poial Fourir Sris, i.. j akig h Fourir rasform, F j F j F F h Fourir rasform of a priodic sigal cosiss of a s of impulss locad a h mulipls of h fudamal frqucy. h wigh of ach impuls is π ims h valu of is corrspodig coffici i h Fourir sris. 4

15 Propris of h Fourir rasform Liariy (Suprposiio) a a a a a, a : arbirary cosas Proof: F j a a j a a a a j a a Liariy of igraio 5

16 Compl Cojuga * * * : Compl cojuga of Proof: F * * j * j * 6

17 Dualiy F F Proof: L L u j d u u ju ju du du hs subsiuios corrspod o a chag bw ad Subsiuig for u j F 7

18 Eampl: Ga fucio rc sa Usig dualiy propry sa rc rc rc Ev fucio 8

19 Proof: j F Coordia Scalig 9 u L u du du du u j u j,, F

20 F F :padd vrsio of () :comprssd vrsio of () :comprssd vrsio of () :padd vrsio of () Eampl:

21 im Shifig j Proof: F j L u u du F ju u du j u ju du j

22 Eampl: Fid h Fourir rasform of -3τ F 3τ rc sa 3 / rc 3 Usig scalig ad im shifig propris, Frc 3 3 Usig liariy, F 6 3 sa 3 / 3 j j 3 j sa 3 sa 3 / rc 3

23 Frqucy Shifig j Proof: F j j j j 3

24 Eampl 5: Fid h Fourir rasform of cos f A rc F A sa j j f cos f f Usig frqucy shifig propris, F A Arc A A rc cos sa sa 4

25 Covoluio Proof: F j d j d j d j d Chagig h ordr of igraio im-shif propry 5

26 Diffriaio j Proof: j d d d j d j d d d - F j j j d d F j 6

27 Igraio j d Proof: F u u u F U d d - U j j Prov i! d u u ui sp fucio τ Shifd vrsio of ui sp fucio τ 7

28 Liar im Ivaria (LI) Sysms LI sysms provid good ad accura modls for a larg class of commuicaio sysms. Som basic compos of rasmirs ad rcivrs, such as filrs, amplifirs ad qualizrs ca b also modld as LI sysms. h impuls rspos impuls. h of a sysm is h sysm s rspos o a ui h y y h h d h frqucy rasfr fucio is h Fourir rasform of h impuls rspos. H F h 8

29 h y h H Y H Usig covoluio propry I im domai, w hav h covoluio igral. O h ohr had, i h frqucy domai, h ipu-oupu rlaio is much simplr, jus giv as a muliplicaio. A LI acs as a filr o h various frqucy compos of h ipu sigal. I migh affc boh is ampliud ad phas. Y Y y j y H H h jh j 9

30 Idal Low Pass Filr (LPF) Idal High Pass Filr (HPF) W W H W H W Badwih=W [rad] Idal Bad Pass Filr (BPF) W W H Badwih=W -W W W 3

31 I pracic, filrs do o hav sharp rasiios as idal os. h badwih is h dfid as h irval of posiiv frqucis ovr which h magiud of rasfr fucio rmais wihi a giv umrical facor. Alhough diffr cririo migh b usd, a widly usd dfiiio is 3dB badwih. Accordig o his criria, h badwih is dfid as h bad of frqucis a which h magiud of h rasfr fucio is a las of is ma. valu. I is calld as 3dB badwih, bcaus rducig h ampliud by a facor of corrspods o a dcras of 3dB o logarihmic scal. H A A 3dB badwih 3

32 Ergy Sigals Dfiiio: h rgy of a sigal () is dfid by E df lim / / Dfiiio: h sigal () is rgy-yp sigal (or rgy sigal) if E is fii. Eampl: Acos E lim A cos lim cos A lim A A 4 si No a rgy sigal! 3

33 Parsval s horm E d : Ergy spcral dsiy * Proof: E * * * d j j d d d Rplac by Ivrs Fourir rasform Chag h ordr of igraio Us h dfiiio of Fourir rasform 33

34 Powr Sigals Som sigals hav ifii rgy, bu hy may hav a fii im-avrag of rgy. his im-avrag of rgy is calld avrag powr. Dfiiio: h avrag powr of a sigal () is giv by P lim / / Dfiiio: h sigal () is powr-yp sigal (or powr sigal) if P A sigal ca o b boh powr- ad rgy-yp, bcaus for rgy-yp sigals P ad for powr-yp sigals A sigal may b ihr rgy-yp or powr-yp. E 34

35 Eampl: Acos E (S prvious ampl) P lim A cos lim cos A lim A A si 4 A Powr sigal a Eampl: u a E a a Ergy sigal P lim a lim a a 35

36 Powr Spcral Dsiy I a aalogy wih h rgy spcral dsiy, w ca dfi powr spcral dsiy (PSD) for powr sigals. P lim / / df S d Drivaio of S () (): powr sigal (): rucad o / ad / rc -/ / 36

37 / / lim / / S d lim d lim Usig Parsval s horm d d akig limi of boh sids Usig PSD dfiiio his rlaioship should hold ovr ach frqucy icrm. 37

38 Dfi h cumulaiv powr spcrum (i.. cumulaiv amou of powr for all frqucy compos blow a giv frqucy ) G df S udu lim u If a irchag i h ordr of limiig opraio ad h igraio is valid, du G S u du lim u du If G is diffriabl, d G S d Udr h abov assumpios S lim 38

39 Powr Spcral Dsiy for Priodic Sigals h prvious discussio o PSD holds for ay gral powr sigal. Now, assum a priodic powr sigal For a priodic sigal, ach priod coais a rplica of h fucio ad h limiig opraio ca b omid as log as is ak as h priod. P / / / / / / m m m * m jm * / * jm / j * / / Eprss i rms of Fourir sris Chagig h ordr of summaio ad igraio j m,, m m 39

40 Fourir rasform Powr Spcral Dsiy S PSD of priodic sigals hav discr compos. I cosiss of a sris of impuls fucios wih wighs corrspodig o h magiud squard of rspciv Fourir sris cofficis. P / / S d 4

41 Priodic Fucio Apriodic Fucio 4

42 Eampl: Acos A A j j j j, : Fourir Sris cofficis - S P S d A h sam rsul was obaid prviously usig im-domai ools. 4

43 H Y Y H H : magiud of h rasfr fucio Ergy spcral dsiy of h oupu sigal Ergy of h oupu sigal Powr spcral dsiy of h oupu sigal Y H E y Y d H S y H S d Powr of h oupu sigal P y S y d H S d 43

44 5 Eampl: A volag sigal is dscribd by u I is applid o h ipu of a idal low-pass filr. h gai of h filr is uiy, h badwih is 5 rad/sc ad h rsisac lvls ar 5 Ω. Calcula h rgy of h ipu sigal ad of h oupu sigal. H H -5 +5,, E E y 3 R R 5 joul 5 3 H 5 d 5 d joul 5 44

45 S F lim - S lim * lim lim lim lim / / / / * * Auocorrlaio Fucio / / * / j j d j d / / u ju u / / ju akig Ivrs Fourir rasform of PSD du ddu j d Rplac by Fourir rasform prssios * u u du lim / / / Chagig h ordr of igraio - * F S lim / / R Auocorrlaio fucio of () 45

46 Eampl ().. - -/4 /4 - A R / * / Sic () is priodic, h limiig opraio ca b omid. R / 4 A / 4 A R (τ) A / R / 4 A A / / / - τ 46

47 Auocorrlaio fucio is widly usd i sigal aalysis. I is spcially usful for h cio or rcogiio of sigals ha ar maskd by addiiv ois. a) Origial sigal c) Nois ) Origial sigal + ois b), d) ad f) : corrspodig auocorrlaio fucios h auocorrlaio fucio of h origial sigal is sill rcogizabl i h oisy cas! 47

48 h y h h u udu R y R * lim y y y lim R / / / / h h * * u udu h v v dv * uh v lim s s u vds dudv * / / * uh vr v ududv h * v h vh vdv * R h h Variabl chag s=-u Dfiiio of R Dfiiio of covoluio 48

49 im-domai h y y R h y * R h h Frqucydomai H Y Y H S y H S 49

50 Symmry R * R Ma-Squar Valu R Propris of Auocorrlaio Fucios * lim lim P Priodiciy / / / / If, i.. () priodic, R R Maimum Valu h auocorrlaio fucio is boudd by is ma squar valu. R R 5

51 R y Cross-Corrlaio Fucio * lim y / / Ry () ad y() ar ucorrlad. Eampl: z y R z? R z y * lim y * R / / R R R y y y 5

52 g af Cross-corrlaio ca b usd o fid h im dlay. ypically usd i sychroizaio for commuicaio sysms. 5

53 Nois Nois cosiss of ay uwad sigals ha d o disurb h rasmissio ad procssig of dsird sigals i commuicaio sysm. Nois may b radom or rmiisic. Nois sourcs: ral --- amosphric ois, ma mad ois, c. iral --- hrmal ois hrmal ois is producd as a rsul of h hrmally cid radom moio of fr lcros i a coducig mdium, such as a rsisor. 53

54 I h cas of radom ois, im-avrag rprsaios ar usful. Ma (DC) valu lim / / Bar idicas im avragig Ma-squar valu (=Powr) P lim / / : roo-masquar (rms) df AC compo Sigal-o-ois raio (SNR) is also widly usd as a prformac masur. S N s sigal ois powr powr S N db s log 54

55 : cosa / lim lim / / / * lim lim / / / * lim / / = = / lim / / lim / / DC powr AC powr 55

56 Bad-Limid Whi Nois A fla powr spcrum coais all frqucy compos wih qual powr wighig ad is dsigad as whi, i a aalogy o whi ligh. S P S d Ifii powr! I gral, h badwih of h rcivr (i.. fro d filr) is arrowr ha h badwih limiaios of h ois procss. If ois has a fla PSD dig byod h badwih of a giv sysm, h ois appars o h sysm as if i wr bad-limid ad whi. S P B B d B B B 56

57 hrmal Nois S h h p k : mpraur [K] k: Bolzma s cosa (=.38-3 joul/k) h: Plack s cosa (= joul-sc) Powr Spcral Dsiy Achivs is maimum a. h valu of his maimum is k. PSD h ra of covrgc is vry slow. S k k h w [rad] k h 6 Hz Byod h opraig frqucy of covioal commuicaio sysms! Coclusio: hrmal ois ca b assumd whi for all pracical purposs. 57