UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EECS 121 FINAL EXAM

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1 Name: UNIVERSIY OF CALIFORNIA College of Engineering Deparmen of Elecrical Engineering and Compuer Sciences Professor David se EECS 121 FINAL EXAM 21 May 1997, 5:00-8:00 p.m. Please wrie answers on blank pages only. Answer all 5 quesions. Clear jusificaions of answers are needed. Problem 1 (20 poins) rue or False Please explain fully. Answers wihou explanaions will ge no marks. a) If E 1 and E 2 are wo evens, hen PE ( 1 E 2 ) PE ( 1 ) PE ( 2 ) only if E 1 and E 2 are independen. b) We have a channel bandlimied o [ W, W], and wan o design a ransmi filer such ha we have conrolled ISI only a he adjacen symbol (no ISI a all oher symbols). I may sill be possible o do his when he symbol rae -- > 2W. 1 c) QAM wih high consellaion size is a suiable modulaion scheme o use for deep-space communicaion. d) Consider he ransmission of binary over a channel wih known ISI, followed by a mached filer and a sampler a symbol rae. One can design a MMSE equalizer in conjuncion wih a symbol-by-symbol deecor ha ouperforms a sequence deecor based on Vierbi s algorihm, in erms of probabiliy of deecion error. e) Non-coheren demodulaion of DPSK (differenial phase-shif keying) resuls in he same probabiliy of deecion error as coheren demodulaion of PSK. 1 of 5

2 Problem 2 (20 poins) Consider an M-ary FSK (frequency-shif keying) modulaion scheme: S m () = 2E s cos[ 2π( f c m )] 0 where is he symbol period, f c is he carrier frequency, and is he frequency separaion. [4 ps.] a) Choose a such ha he signals are all orhogonal. Verify ha hey are. [6 ps.] b) Design an opimal coheren demodulaion and deecion scheme, assuming perfec phase esimaes. [5 ps.] c) Derive an expression for he probabiliy of deecion error. [5 ps.] d) Derive a union bound for he probabiliy of error. 2 of 5

3 Problem 3 (20 poins) Consider a channel wih ISI, wih impulse response: h() = δ () δ --. ransmission is done via binary 2 wih ransmi filer g (). he receiver is composed of a mached filer followed by a symbol-rae equalizer followed by a symbol-by-symbol deecor. { } g () h () Mached filer Symbol-rae equalizer deecor he symbols in he informaion sequence { } are assumed o be independen and equally likely o be 1 or 1. [10 ps.] a) Design a zero-forcing linear equalizer o cancel he effecs of he ISI. Does i depend on he saisics of he informaion sequence? [10 ps.] b) Design a 2-ap MMSE linear equalizer. Does i depend on he saisics of he informaion sequence? E ---- s g () 3 of 5

4 Problem 4 (20 poins) In class we sudied he Miller code which is an example of a run lengh-limied code. In his problem, we will look a anoher code, which is an example of an error-correcing convoluional code. he informaional sequence { } is a sequence of equally likely 0 s and 1 s. he coded sequence { b m } is given by b 2n = 2 b 2n 1 = 1 n = 012,,, where is addiion mod 2. he symbol sequence { b m } is ransmied over an AWGN channel using anipodal 1 a symbol rae --. { } Coder { b m } binary rae = 1 [3 ps.] a) Consider as an example he informaion sequence ({ }) { 01100,,,, }. Find he corresponding coded sequence { }. (Assume ha a k = 0 for k < 0 ). Wha is he lengh of he coded sequence? b m [3 ps.] b) Wha is he rae a which informaion bis is ransmied over he channel? We now wan o design a deecor for he informaion sequence { }. [4 ps.] c) Recall ha for he Miller code, he previous informaion bi serves as he sae of he sysem. Wha can be used as he sae for his problem? How large is he sae space? Enumerae all possible saes. [5 ps.] d) Draw he rellis, wih each sage of he rellis corresponding o an informaion bi ransmied. [5 ps.] e) Using he rellis or oherwise, describe as clearly as possible how opimal deecion of he informaion sequence { } can be done a he receiver. 4 of 5

5 Problem 5 (20 poins) Consider a muliple-access communicaion sysem, wih wo ransmiers and one receiver. Boh ransmiers use binary. Le g () be a recangular ransmi pulse. Sender A uses he pulse g (), and sender B uses he pulse g ( ). A ime 0, sender A ransmi symbol X and sender B ransmis symbol Y, where X and Y are independen and equally likely o be 1 or 1. he overall ransmied signal is: U () = Xg () Yg ( ). Sender A Sender B Xg () Yg ( ) U () Receiver g () E ---- s 0 ( Xˆ, Ŷ) he received signal is U () where { } is independen AWGN wih power specral densiy [7 ps.] a) Design a maximum likelihood receiver for esimaing he pair of ransmied symbols ( X,. N 0 Now suppose due o problems of synchronizaion beween he wo senders, he ransmied pulses overlap. Specifically, he overall ransmied signal is now U () = Xg () Yg ( α) where 0 < α < 1 gives he fracion of ime of overlap. [6 ps.] b) Assuming ha he receiver mainains perfec synchronizaion wih boh senders, we plan o use he following receiver srucure: g () X Y and decide X = 1 if R A 0, X = 1 if R A < 0. Similarly we decide Y = 1 if R B 0 and Y = 1 if R B < 0. Compue he probabiliies of deecion error for boh X and Y (i.e., P( Xˆ X) and P( Ŷ ). Xg () Yg ( α) U () g ( α) 2 α [7 ps.] c) Does he above scheme minimize he probabiliies of deecion error of he pair ( X, (i.e., P( ( Xˆ, Ŷ) ( X, )? Explain. If no, design one ha does. (Hin: Are he signal waveforms of he wo senders orhogonal?) 5 of 5 0 α R A R B