$($a$)$ Refer toTDOAlocalizationinathree$-$dimensionalspace. Assumethatfivereferencenodesare

knownatNode$1$:$(0,3,0),$Node$2$:$(6,0,0),$Node$3$:$(3,4,0),$Node$4$:$(4,3,0),$andNode$5$:$(0,0,8)$

respectively.Also,fornode U$,$ weknowthefollowingpropagationtimes:

U$11=0$s$,$U$12=1$s$,$U$13=0\mathrm{.}7$s$,$U$14=0\mathrm{.}7$s$,$& U$15=1\mathrm{.}7$s$.$ The velocityofpropagationis v$.$

$($i$)$ Findtheunknownlocation$($xU$,$ yU$,$ zU$).$

$($ii$)$ Nowassumethatthepropagationspeedisknowntobe$8\mathrm{.}7$m$/$s$.$Findtheunknownlocation

$($xU$,$ yU$,$ zU$).$

$($b$)$ NodeAsendsasynchronizationrequesttonodeBat$3150($onnodeA$$sclock$).$At$3250,$nodeA

receivesthereplyfromnodeBwithatimestampof$3120.$

$($i$)$ WhatisnodeA$$sclockoffsetwithrespecttothetimeatnodeB$($youcanignoreanyprocessing

delaysateithernode$)?$

$($ii$)$ IsnodeA$$sclockgoingtooslowortoofast$?$

$($iii$)$ HowshouldnodeAadjustitsclock?

$2$

$($c$)$ TwonodesAandBuseRBStoreceiveperiodicacousticsynchronizationsignalsfromareferencenode.

NodeA$$sclockshows$10$swhenitreceivesthelastsynchronizationbeacon,whilenodeB$$sclockshows$15$s$.$

NodeAdetectsaneventattime$15$s$,$whilenodeBdetectsthesameeventattime$19\mathrm{.}5$s$.$Assumethat

nodeAis$100$mawayfromthesynchronizationsourceandnodeBis$400$mawayfromthesynchronization

source. Whichnodedetectedtheeventsoonerandbyhowmuch?Assumeasignalspeedof$300$m$/$s$.$

$($d$)$ TOArequiresthatallthereferencenodesandthereceiverhaveprecisesynchronizedclocksandthe

transmitted signalsbelabeledwithtimestamps.TDOAmeasurementsremovetherequirementofan

accurate clockatthereceiver.Assumethatfivereferencenodeshaveknownpositions

$(0,0),(1,1),(0,1),(3,1),$ and$(1,4)$ respectively.Wechoose$(0,0)$ asthereferencesensorfordifferential

time$-$delayswhicharedefinedas

t$1$r $=$ t$1$ tr $=$

rs$1$ rs$2$

v

where v is thevelocityofpropagation, rsi is thedistancebetweentheunknownnodeandthe i th node.

Furtherassumethat t$12=1\mathrm{.}4$s$,$ t$13=0\mathrm{.}4$s$,$ t$14=1\mathrm{.}6$s$,$ and t$15=2\mathrm{.}6$s

$($a$)$ Findtheunknownlocation$($xt$,$ yt$).$

$($b$)$ Nowassumethatthepropagationspeedisknownas$1\mathrm{.}8$ m$/$s$.$ Findtheunknownlocation$($xt$,$ yt$).$