Hw 1 Solutions

Homework Assignment Assignment 1: Physical Layer 18759: Wireless Networks [Ma! 1"" #oints$ %&e %ate:8th 'e()*n class+

Pro(lem 1: ,hannon -a#acity .heorem: [15 #oints$ According to the Shannon’s Channel capacity theorem: Channel capacity C = B*log (1 + S/N) Where B = Bandwidth and S/N = Signal to Noise Ratio. Answer the following questions with respect to the Channel c apacity theorem: (1) When the Bandwidth increases what happens! (") When the Signal to #oise $atio increases what happens! (%) When the Signal to #oise $ation S&# tends to infinity what ha ppens! (') When the Bandwidth tends to infinity what happens! Solution: (1) When the Bandwidth increases what ha!!ens" As the andwidth of the channel increases it is possile to mae faster changes in the information signal therey increasing the information rate and hence the channel capacity cap acity** (#) When the Signal to Noise Ratio increases what ha!!ens" As S&# increases one can increase the information rate while still pre+enting errors due to noise* ($) When the Signal to Noise Ration S/N tends to in%init& what ha!!ens" As S&# ,- infinity an infinite information rate is theoretically po ssile irrespecti+e of the andwidth of the channel* chann el* (') When the Bandwidth tends to in%init& what ha!!ens" As Bandwidth ,- infinity infinity the channel capacity cap acity does not approach infinity since with an increase in Bandwidth the noise n oise power also increases*

Pro(lem /: ,hannon an0 Ny&ist e&ations [/" #oints$ Suppose that the spectrum of a channel is etween 1. /0 and 1" /h and an intended capacity of 2 /ps* (1) What should e the S#$ in order to otain this capacity! (") 0ow many signaling le+els are required to otain this capacity! (%) What would e the capacity if the en+ironment en +ironment starts suffering suffering lesser noise and the S#$ goes up to "3 dB* (') Same question question as (") ut for for the capacity capacity in (%)

Solution: B4" /h4"51.67 C42 /ps4251.67 ps (1) What should e the SNR in order to otain this ca!acit&" Shannon8s 9heorem: C4B5log"(1S#$) ;4- "6(C&B),14S#$;4- S#$41< (#) ow an& signaling leels are re,uired to otain this ca!acit&"  #yquist 9heorem: C4"B5log"(/) ;4- "6(C&"B)4/ ;4- /4' ($) What would e the ca!acit& i% the enironent starts su%%ering lesser noise and the SNR goes u! to #- dB. S#$(dB)41.5log1.(S#$);4-S#$41.6"*3;4-S#$4realistic? solution*

Pro(lem 2: ,hort Answer 3&estions [/5 #oints$ (1) What do you mean y @ynamic equaliation and what prolem does it tacle! (") Why is it that when you increase the da ta rate of a communications system typically the error rate increases! (%) =plain C or forward error correction scheme with an applicale e=ample* (') What are the ad+antages and disad+antages of Spread Spectrum techniques! (n odeling a gien wireless counications s&stes in which the two node are in an uran scenario with no ?@S !ath 4dditie White 7aussian Noise channel odel is a good guide %or the designer. 3alse:  AW# model considers thermal noise ut does not consder the effects of multipath and the lac of ENS path* Dn this scenario the $ayleigh channel model would e appropriate* ($) 3re,uenc& iersit& is achieed & s!reading the signal in tie & sending ulti!le co!ies o% the signal at di%%erent tie instants. 3alse: 9ime di+ersity is e=ploited y sending multiple copies of the signal at different time instantsG frequency di+ersity is used y sending the signal at different frequencies* (') ulti!ath !ro!agation alwa&s i!roes the SNR. 3alse: /ulti,path effect can attenuate the signal recei+ed at the recei+er if the multiple copies recei+ed are out of phase* 0owe+er it can also cause constructi+e interference at the recei+er if the signals recei+ed are in phase hence causing the signal to increase in strength*