AE57/AC51/AT57 SIGNALS AND SYSTEMS DECEMBER 2012

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1 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Q. Drmi powr d rgy of h followig igl j i ii =A co iii = Solio: i E P I I l jw l I d jw d d Powr i fii, i i powr igl ii =A cow E P I co w d / co l I I l d wd d Powr i fii, i i powr igl co wd IEE

2 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER iii = I I N N N N N E P l I N l N I N Powr i fii, i i powr igl Q b Giv how i Fig. Skch h followig i - ii - iii iv -+ Solio: i - ii - IEE

3 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER iii iv -+ Q Drmi h Forir Sri rprio for igl; i i / ii co / 6 8 Solio: i i / i priodic wih priod N = 4 Uig Elr forml j j / j / j / j j j j / j k k j k Comprig wih DFS qio j k jj j jj / / k k k k k, IEE

4 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER ii co / 6 8 i priodic wih priod N = 6 Uig Elr forml j 8 j / 6 j / 6 j / j 8 j / 6 k k j k Q b Comprig wih DFS qio jj / 6 k k k jj / 6 k 7 k 8 k, S d prov h followig Forir ri propri of coio priodic igl. i Frqcy hif propry ii Sclig propry Solio: i Frqcy hif Propry bl.. Pg No. 6 of Book - I ii Sclig propry If i priodic igl h f= i lo priodic. If h fdml priod h f h fdml priod / If h I. Forir ri coffici of d r idicl Proof: ic f9 h fdml priod / IEE 4

5 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER F f jkw d jkw F d P p= h =p/ d d=/dp jkw p F p dp jkw p p dp F F If h i. Q4 S d prov Prvl rgy horm for coio igl. priodic Solio: Sm: h rgy my b fod from h im igl or i pcrm j i. E d j d Proof: Ergy of igl i giv by E d * d h Forir rform d i ivr i j j d j j d kig cojg for h bov qio *j * * j d *j j d Sbi i qio IEE

6 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER E E j jd Uig qio E E j d *d * j *j d j d j d hi rlio i clld Prvl horm or Ryligh rgy horm. Q4 b h rfr fcio of h ym i giv by: j Hj j j Fid h ym qio d lo impl rpo of h ym. Solio: H jw jw Y jw jw jw jw jw jw jw jw jw Y jw jw jw kig IF d y d y d y d d d jw H jw jw jw L m=jw m A B H jw m m m m Solvig A= d B=- IEE 6

7 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Q H jw m m H jw jw jw hig IF i g jw rlio h S d prov h followig propri of dicr im Forir rform. i im hifig propry ii Diffriio i frqcy domi Solio: i im hifig propry: Sm: F If j h F j j Shif i im domi will rl i mliplyig by poil i frqcy domi Proof. F { } L d d d j j d d j j ii Diffriio i im domi propry: If F j d d h jw j d Diffriig igl i im domi i m mliplyig hir pcrm IEE 7

8 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Q b i frqcy domi Proof: Ivr F jw jw Diffriig wih rpc o d jw jw jw d From h bov qio w hv d jw j d Coidr dicr im LI Sym wih impl rpo. h = whr <. U Forir rform o drmi h rpo o h ip = β wih β < Solio: Q6 Empl., Pg o. 8 of Book - I Drmi h Nyqi r for h followig igl i =+coπ+4i4π ii =co6π co8π Solio: i = +coπ+4i4π f = H d f = H f Nyq =fm m ==4H ii = co6π co8π =co4 π+co π Q6 b f =7 H d f = H f Nyq =fm m =7=4H Wih digrm pli mplig of dicr im igl. Solio: Smplig horm. Sm: L m i mg igl bd limid o f m H, if hi igl i IEE 8

9 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER mpld r f f h w c rcorc h mg igl from h m mpld vl wih miimm diorio. i. f f m whr f i mplig frqcy d fm i mimm mg frqcy L m=mg igl m M f i priodic dl fcio wih Forir ri f f f Smpld igl S=m Mliplicio i im domi i m covolio i frqcy domi S f M f * f M f * f f Covolvig y fcio wih dl fcio yild h m fcio S f M f f Spcrm of mpld igl i priodic wih priod f. Q6 c Fid h frqcy rpo d impl rpo of h ym wih ip = - d op y= -. Solio: Applyig F for ip d op igl F{ } y F{ y } Y jw Frqcyr k ig H jw h IF jw Y jw jw rpo jw jw jw jw jw IEE 9

10 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER IEE Q7 Fid, h op y of h ym dcribd by h diffril qio y d dy by Lplc rform mhod. Am h h ip = - d iiil codiio i y + = -. Solio: y d dy = -, y + =- kig Lplc rform y kigivr L Y Y B A B A Y Y y Y Q7 b Uig covolio propry. from Fid S Solio: S Covolio propry of L i

11 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER IEE, * for d w w Q7 c Fid h ivr Lplc rform of 8 4 Solio: Uig h rlio i i i IL b b Q8 i Fid h Z-rform of h followig qc d fid h ROC i i ii Solio: i co i i ROC Uig clig propry

12 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER IEE ii 9 co / i {/ i ROC Uig hifig propry 9 co / i {/ i Z 9 co / i {/ i ROC. ROC: d /, Roc : / Q8b i S d prov i Iiil vl horm of -rform ii im Epio propry of -rform Solio: Iiil vl horm: Sm: If i cl d Z l h Proof: By dfiiio

13 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER For cl kig limi l o boh id l l Q8 b ii Q9 Pg o. 769 o 77 of Book I Dfi h followig rm wih rfr o probbiliy hory i Smpl pc ii Ev iii Mlly cliv v iv Codiiol probbiliy v Joi probbiliy vi Powr pcrl diy Solio: Smpl pc: S coi of ll poibl ocom of prim Ev: Ev i b of mpl pc Mlly cliv v: If wo v r mlly cliv h hr i o commo lm bw hm. Codiiol probbiliy: IEE

14 AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER Probbiliy of v dpd o om ohr v PA/B-probbiliy of v A fr h v B i ovr. Joi Probbiliy: PAB=PAPB/A if A d B r iiclly idpd h, PAB=PAPB h powr pcrl diy: PSD, dcrib how h powr or vric of im ri i diribd wih frqcy. Mhmiclly, i i dfid h Forir rform of h ocorrlio qc of h im ri Q9 b Wh i wid iory proc mio i propri. Solio: A rdom proc i clld wid iory if i ifi Q9 c. M of h proc i co. ocorrlio fcio i idpd of im. vric of h proc i co Wri hor o o: i Gi proc ii Ergodic proc Solio: i Gi proc - Pg o. 4 o 8 of Book - II ii Ergodic Proc - Pg o. 4 o 4 of Book II E BOOKS. Sigl d Sym, A.V. Opphim d A.S. Willky wih S. H. Nwb, Scod Ediio, PHI Priv limid, 6. Commicio Sym, Simo Hyki, 4h Ediio, Wily Sd Ediio, 7h Rpri 7 IEE 4

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%