Solution Set #1

Transcription

1 Soluton Set #. Fnd epressons and setch the results of the followng operatons: (a) COMB RECT The spacng of the elements of the COMB functon matches the wdth of the rectangle; we can do ths n ether doman easly, but I ll to the frequency doman frst: n n o n F COMB RECT F COMB F RECT o o COMB [ ] SINC [ ]! () δ [ ] SINC [ ] COMB RECT () () () () () + + µ δ δ () 0 () δ [] F {() δ []} [] SINC [ ] SINC [ ] SINC [ ] SINC SINC [] (unts of delta ) g [] (red sold) and f [] SINC (blue dashed), showng the the nput IS recovered correctly.

2 (b) COMB TRI F n COMB TRI o COMB TRI F n n COMB F TRI o o COMB [ ] SINC [ ]! () δ [ ] SINC [ ] () () () () () + + µ δ δ () 0 () δ [] F {() δ []} [] SINC [ ] SINC [ ] SINC [ ] SINC SINC [] (unts of delta ) g [] (red sold) and f [] SINC (blue dashed), showng the the nput IS recovered correctly.

3 (c) COMB SINC F n COMB SINC o COMB RECT n n F COMB F SINC o o COMB [ ] RECT [ ]! () δ [ ] RECT [ ] () () () () () + + µ δ δ () 0 () δ [] F {() δ []} [] RECT [ ] RECT [ ] RECT [ ] RECT RECT [] (unts of delta ) g [] (red sold) and f [] SINC (blue dashed), showng the the nput IS recovered correctly. 3

4 (d) COMB SINC n F COMB SINC h o COMB SINC h n n F COMB F SINC o h o COMB [ ] TRI[ ]! () δ [ ] TRI[ ] () () () () () + + µ δ δ () 0 () δ [] F {() δ []} [] TRI[ ] TRI[ ] TRI[ ] TRI TRI[] (unts of delta ) g [] (red sold) and f [] SINC (blue dashed), showng the the nput IS recovered correctly. 4

5 . For s [] COMB [], h [] SINC [], andf n [] SINC (n,, 3), fnd the forms of and n setch the three functons g n [] (f n [] s []) h [] Soluton: (straght off of the fnal eam) ³ h g [] (f [] s []) h [] SINC COMB [] SINC [] SINC [] COMB [] SINC [] + n + n δ [ n] SINC [n] δ [ n] + n SINC [] δ [ n] + SINC [ ] δ [ ( )] + SINC [0] δ [ 0] + SINC [+] δ [ (+)] + δ [] g [] (f [] s []) h [] δ [] SINC [] SINC [] g [] SINC [] 6 SINC [] f [] y g [] (red sold) and f [] SINC [] (blue dashed), showng the the nput s not recovered correctly. ³ h g [] (f [] s []) h [] SINC h SINC + COMB [] + n n SINC h n COMB [] SINC [] SINC δ [ n] δ [ n] + SINC δ [ ( )] + SINC [0] δ [ 0] + SINC + + n h! SINC n δ [ n] SINC [] δ [ (+)] + 5

6 whch sn t much help, so let s evaluate n the frequency doman Setch them: G [] (F {f []} F {s []}) F {h []} (F [] S []) H [] n h F [] F {f []} F SINC o TRI[] TRI S [] F {s []} F {COMB []} COMB [] H [] F {h []} RECT [] µ G [] TRI COMB [] RECT [] h (a) F [] S [] TRI (F [] S []) RECT [] TRI COMB [], (b)rect [] (blue) and n o (red); (c) F TRI h h SINC whch shows that G [] TRI g [] SINC f [] y g [] (red sold) and f [] SINC (blue dashed), showng the the nput IS recovered correctly. 6

7 Graph t: ³ h g 3 [] (f 3 [] s []) h [] SINC 3 F 3 [] F {f 3 []} 3 TRI[3] 3 TRI S [] F {COMB []} COMB [] " # F 3 [] COMB [] 3 TRI COMB [] 3 COMB [] " # 3 SINC [] F 3 [] S [] (red) and (F 3 [] S []) H [] (blue) From the graph, we see that the transfer functon cuts out the central replca of the trangle from the sawtooth spectrum. Thus the mage g 3 [] s: g 3 [] SINC f 3 [] 3 y g [] (red sold) and f 3 [] SINC 3 (blue dashed), showng the the nput IS recovered correctly. Note that f [] and f 3 [] were correctly maged by the system, but that f [] s not. 7

8 3. A samplng functon s [] s used to sample a sgnal f [] as follows: f s [] f [] s []. Thesampled functon f s [] s then passed through a lnear shft-nvarant system wth transfer functon H [] to produce an output g []. For: f [] SINC [00] s [] s COMB [ s ] H [] RECT (a) Fnd the mnmum samplng rate Nyqust (Nyqust rate) that wll permt eact recovery of f [] from f s [];.e., such that g [] α f [] (α s a constant). F [] F SINC [00] ª 00 TRI 00 ma 00 cycles unt Nyqust ma 00 cycles unt s F s [] F{f [] SAMP []} 00 TRI COMB 00 s 00 TRI COMB G s [] F s [] H [] µ 00 TRI TRI 00 g s [] SINC [00] (b) Fnd g [] when s Nyqust,andsetch. COMB 00 δ [] 00 TRI s Nyqust 00 cycles unt f s [] SINC [00] (00 COMB [00]) F s [] 00 TRI 00 COMB 00 TRI as seen n graph + n TRI δ [ 00n] RECT s n + n δ 00 n 00n TRI [] 00 8

9 G s [] [] RECT RECT h g s [] 00SINC [00] 00 SINC 00 9

10 4. As n the prevous problem, f s [] f [] s []. Wth: f [] SINC [0] s [] (RECT [] s COMB [ s ]) RECT [00] H [] RECT s (a) Fnd g [] f s s chosen to be the usual Nyqust rate assocated wth a COMB samplng functon of nfnte etent. Soluton: Ths problem s meant to llustrate the dffculty wth the Gasll epresson for realstc samplng. What does t mean to Frst calculate the epressons n the frequency doman: F{f []} F [] 0 RECT Nyqust 0 cycles ³ s [] RECT 0 COMB [0] 5 µ S [] F{s []} 5 SINC [5] COMB 0 unt RECT [00] SINC F [] S [] µµ 0 RECT 5 SINC [5] COMB SINC The last SINC functon 00 SINC s wde compared to the other functons 00 µ µ F [] S [] ' 0 RECT 5SINC [5] COMB SINC 00 µ 0 RECT COMB 5SINC [5] SINC 00 µ 0 RECT COMB 5SINC [5] SINC 00 µ ' 0 RECT COMB SINC 00 The narrow SINC functon creates rpple n the convoluton, but of small ampltude: G [] (F [] F{SAMP []}) H [] µµ ' 0 RECT COMB SINC RECT 00 0 ' 0 RECT RECT RECT 0 ½ ¾ g [] 'F 0 RECT SINC [0] SINC 0 0, but somewhat smoother because the ampltude near ' 5 s somewhat attenuated (b) Fnd g [] f s s chosen to be twce the Nyqust rate determned n part a. s Nyqust 0 cycles unt F [] S [] µµ 0 RECT 5SINC [5] COMB 0 µ ' 0 RECT COMB SINC 00 µ ' 0 RECT COMB SINC 00 G [] g [] agan wll closely reproduce f [] 0 0 RECT SINC 00 0

11 (c) What parameter changes wll mprove sgnal recovery? Sgnal recovery s enhanced by ncreasng the wdth of RECT 5 h and decreasng the wdth of RECT 00 n s [] (d) Can f [] ever be recovered eactly n ths case? NO!

12 5. Consder the followng sequence of operatons appled to the nput functon f []: g [] ((f [] p []) s []) h [] f [] SINC [0] p [] 00 RECT [00] s [] RECT 5 s COMB [ s ] H [] RECT s (a) Fnd g [] f s s the usual Nyqust rate for deal samplng wth a COMB functon of nfnte etent. f [] SINC [0] F [] 0 RECT cycles 0 ma 5 unt length p [] 00 RECT [00] P [] SINC p [] blurs f [] slghtly 00 s [] RECT 5 s COMB [ s ] S [] 5 SINC [5] COMB s H [] RECT h [] SINC s s s NY Q cycles ma NY Q 0 unt length ma 0 unt G [] ((F [] P []) S []) H [] µµ 0 RECT SINC (5 SINC [5]) COMB RECT µµ 0 RECT [] (5 SINC [5]) COMB RECT 0 0 0!! + 0 RECT (5 SINC [5]) 0 δ [ 0 ] RECT 0 0!! RECT (5 SINC [5]) δ [ 0 ] RECT RECT SINC [5] 0 s 0 f [] p [] SINC [0] 00 RECT [00] SINC [0] wth slght blurrng g [] 50 SINC [0] 5 RECT 0 SINC [0] RECT 5 5 (b) Fnd g [] f s s twce the rate used n part (a). cycles ma ma 5 unt length cycles NY Q 0 unt length 0 unt

13 µµ G [] 0 RECT SINC (5 SINC [5]) COMB !! 0 RECT (5 SINC [5]) δ [ 0 ] RECT 0 0 RECT (5 SINC [5]) 0 g [] 0 SINC [0] RECT (wndowedsincfuncton) 5 RECT 0 0 (c) What parameter changes wll mprove the fdelty of g [] compared to f []? ncrease s,maep[] narrower whle retanng unt area (d) Do choces of the parameters est that allow f [] to be recovered eactly? fnte support of s [] and nfnte support of f [] g [] s always alased, so cannot be recovered eactly. 3