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
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
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
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 / 6 8 8 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
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
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
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
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
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
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
AE7/AC/A7 SIGNALS AND SYSEMS DECEMBER IEE, * for d w w Q7 c Fid h ivr Lplc rform of 8 4 Solio: 4 8 4 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
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
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
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