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Chi nese Journal   of  El ectroni cs   Vo1 . 14,No. 4,O ct.2005 

i m e D el ay and 

D oppl er S  hi f t Est  i m at   i on术   ZHANG  W ei qi ang and TAO Ran 

( De pa r t me nt   o f   El e c t r o ni c   Eng i ne e ri ng,Be qi ng   I ns t i t ut e   o f   Te c h no l o g y,Be i j i ng   10 0081 ,Ch i na )   A bstract—— Several  C ram er-R ao l ow er bounds for tim e 

The  r es t   of   the  paper   i s  organi zed  aS  f ol l ows.The  m odel i ng 

delay and D oppler shift estim ati on have been obtained f o r 

and  probl em   statem ent   are  des cri bed  i n Secti on  II .In  Secti ons  

di ff. erent m odels by using various m ethods, but the re- 

I II,I V and V,the  deri vati ons  of  CRLBs  f or   j oi nt  TD and DS 

searches are not system ati c and extensive. In this PaPer,   starti ng w i th a general  m odel,C ram er-Pato low er bounds  o r ti f m e delay estim ati on,D oppl er shi ft estim ati on, and 

joi nt  ti m e del ay and Doppl er  shi f t  esti m ati on are deri ved  by using F i sher i nf orm ati on m atri x. T he present results 

es ti mati on,TD es ti m ati on and DS es ti mati on  ar e presented.   Secti on  VI  of iers  t he r el ati ons  between t he  CRLBs  and som e  

f ur ther   com m ent s.Se cti on  VII   pr ovi de s  som e  s i mul ati ons.Fi .   nal l y.i n  Se cti on VI II   the  concl usi on i s  gi ven.  

are consi stent w ith the previous ones;m oreover,they a r e  applicable to all  si gna1 .to-noise rati os. T he bounds ob-  tained ander di   rent condi ti ons are com pared and the  relati ons betw een them  are off. ered. Further explanations  about the relati ons are also gi ven from  the poi nt of  view  of  inform ati on theory. Sim ulati ons are provided to support 

II.M odel i ng and Probl em  Statem ent  Le t   a ss ume   t ha t   t he   s our c e   s i gnal   s( t )i s   nar r o w— ban d  and  the  echo si gnal   wi l l   be  

the theoretical  results.  

r( t )=as( t—T) e j (  d 蚪 )+n( t )  

( 1 )  

K ey w ords —— C ram er-R ao low er bound,T im e del ay,  

where  a i s  a r eal  at tenuati on  cons tant,  

D oppler shif t,Estim ation.  

i s  a  cons tant   pha s e,  

and r and  d  denote ti m e del ay and Doppl er  shi f t,r espec —  

I.Introducti on 

t i v el y.n( t )i s   t he   er godi c,z e r o - me an, c ompl e x  whi t e   Ga ss n i an  noi s e .We   ss a ume   t hat   t he   e ner gy   of   t he   e c ho  s i gnal( wi t ho ut   n oi s e )i s  

In  radar,sonar,and  other   appl i cati ons,i t   i s   of ten  requi r ed 

t o  e s t i ma t e  Ti me   de l ay( TD)a nd/ or  Doppl e r  s hi f t( DS)of   t he   e c ho  s i gnal   r e le f c t e d  by  mo vi ng/ s t a t i c   t ar ge t s 【   J .The   Cr a me r . Ra o  l o we r   boun d f CRLB)f o r  TD  an d/ or   DS  i s   US U—  

E=0  /   。 。   I s ( t )   l d t   一

。 。 

and the  power  s pectr al   densi t y  of   the  noi se  i s  N o,that  i s  

E{ n( t ) n( t , ) )=0   E{ n( t ) n ( t , ) )=No S( t—t   )  

al l y an i nter es ti ng probl em  because  i t  can gi ve the f unda-   m ental  l ower  bound f or  the err or  var i ance  of the  est i m ate 

( 3)   ( 4)  

Daram eters[ 6, 7 1 . 

Ther e are  several   resul ts  obtai ned f or  di ierent f  model s  by 

us i ng  v ar i ous   me t hods .I n   Re f . [ 7 】 ,Ka y  go t   t he   CRLB  or f   TD  e s t i ma t i o n.I n  Re f . 『 81 ,Chan g  obt a i ne d  t he   CRLB  or f   TD  a nd  DS  e s t i ma t i on.I n  Ref . 『 9 1 ,Li n  and   Ke   de r i ve d  t he   CRLB  or f  

whe r e   E{・ )i s   t he   e xpe c t a t i on  o pe r a t or ,  ( ・ )i s   t he   Di r ac   de l t a  f unc t i on.and ‘ ‘  ”de not e s   t he   c ompl e x   c on j uga t i on.  

III.CR LB f or Joi nt  TD and D S  Esti m ati on 

i oi nt   TD and DS  es ti mati on  us i ng  the  irs f t— order   per turbati on  approxi m ati on method,but  the resul ts  onl y can be  us ed f or  

I f   we  have no other  i nf orm ati on,the  unknown vect or  wi l l  

hi gh   Si g na l - t o - noi s e   r at i o( SNR) .Ot he r   e s t i ma t i o n  pr obl e ms,   such  as  that  rel ated  to mul ti pl e  sour ces,mul ti path  and  wi de -   band  si gnal s  have  al so  been s tudi edt  ”一 

0=[ 0  

r   d】 T  

( 5)  

. 

In thi s pape r,we start  wi th a gener al  m odel  and deri ve 

The  c on di t i onal   Pr o babi l i t y  dens i t y f un ct i ons( PDF)of   t he   echo  si gnal  i s  gi ven by 

t he  CRLB f or  TD e s t i mat i on,DS  e s t i ma t i on,and   j o i nt  TD 

an d  DS   e s t i mat i on   by  us i n g  Fi s her   i nf or ma t i on  ma t r i x( FI M) .   We  al s o di scus s the r el ati ons  bet ween  the m  and gi ve  f ur ther  



K   e x p { 一 志  c ) _ 0 s ( t 一   ) e j ( w d t +  ̄ ) I   d   J  

expl anati ons.   M anuscri pt Recei ved Jul y 2004;Accepted M ar.2005.Thi s work i s supported by the N ati onal  Natural  Sci ence Foundati on of China 

f No . 6 02 3 20 1 0) ,b y   t h e   Te a c hi ng   a nd   Re s e a r c h  Awa r d   Pr o gr a m  f or   Out s t a ndi ng   You ng   Te a c h e r s   i n   Hi g he r   Ed uc a t i o n   I n s t i t ut i on s   o f   MOE,   China.  

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636  

Chi ne s e   Jo ur nal   o |El e c t r oni c s  

200 5  

=  l np 

一 

l n  

)  

o o 

_,3l一

m   

l s ( ~

Re  

 

。  

( 24)  

s 

。 

( 2 5)  

Taki ng parti al   deri vati ves,we  can  get   =  =

Re  

 

( …

 

卅  t  

( 8)  

m 

…  ̄ ̄

_,32= 一  2a2   nn

s * ( t - v ) h ( t - v ) d t =  Ⅳ 砌2 7 )  



= 





m 

( …

 

卅  t

( 9)  

z  z  瓦 2 a 2   R e f

Re  

 

( 2 6)  

 



r  



:2sⅣ聊

  ( 2 8)  

。 

/3 4=143  

= 

m 

州 啪 

2a2

e  



whe r e  ( t )=ds( t ) / dt ,a nd  Re {・ )and  I m{・ )de not e   r e al   pa r t   and i magi nar y  part  of   a  f unct i on,res pecti vel y.  

 

I m 

( …

  …

) dt  

=一2 SNR{  ̄ - - i+07 - - )  

( 29)  

Ther ef ore,the  FI M  becomes  

It   i s  wel l - known  that   the  CRLB  of   any  unbi ase d es ti mat or  must  s ati sf y[ 。]  

r1 / 口2  

I ( O ) = 2 S N R [  一 0 1 面    —  0 面   2  

0 

1   0 

v ar ( O ̄ )  [ r (  ) 】  

( 1 2)  

t+ 丁  一  t一 西丁 

whe r e【 1-1 ( 9) 】  de n ot e s   t he【 i , i 】e l eme nt   of   t he   i nve r s e   of   t he   FI M  J( 9) ,whi c h  i s   de ine f d  by 

t+ 1-   一

 

t一 0  7-  

( 3。 )   Inverti ng  the  m atri x  yi el ds   the  CRLBs  f or  TD and DS 

[  )  =  :E {   0 l n a p   ( r ; 0 ) 0 l n 0 p 0 ( 5 r   ; 0 )  

(  3)  

Va r ( I  l T R    T   B1:  l 』  I (   口J l   3 3: 2SⅣ R B262一 a 

Bef or e  deri vi ng  t he  FI M  el em ents,we   irs f t   def ine   s om e   symbol s  

whi c h  wi l l   be   US e d  l a t e r ・  



2SⅣ R 口2  2一 a0  

譬厂 。 。  ㈣  t   厂 。 。  s ∽  t   譬   m厂 。 。 s   t   譬厂 。 。 o c圳。      

西 =



( 31 )  

var (  ) cRLB1=【 r ( 9) 】 4 4=  ( 1 4 )  



]  

+ 2r  + r。 

( 32 )  

( 1 5)   whe r e  

a=瓦 一0 

。= — 0 2 — 2一— 0 2 — 2 ,   。=  一 

( 33)  

( 1 6)   W e can noti ce  that  t he  CRLBs  are  consi stent  wi th t he 

( 17 )  

resul t s  obtai ned  thr ough  t he  f irs t. orde r  pert urbati on  appr oxi -   m ati on  m et hod[ 9, PP. 8 1   JI n  f a c t, i n  Re f . [ 91 ,t he   Ma xi mum  l i ke l i -   .

( 18 )  

hood  fML)esti mat or   i S  us ed,whi ch can achi eve  t he  CRLB  at   hi gh SNR.Ther ef ore,the  consi stenci es   are  r ea s onabl e.But  i n  theory,the  perturbati on  approxi m ati on  m ethod  cannot  appl y 

譬   m   ㈣  



( 1 9)  

to l OW  SNR .whi l e  our   m ethod  can  be  used  at  al l  SNRs.  

These  symbol s   have  real   physi cal   meani ngs  and  can  al so  be  ex-  

I V .CR LB f 0r T D E sti m ati on 

press ed i n f requency  dom ai n[ 引 but  we  do not  expati ate  her e.   .

Usi ng s om e r outi ne  deri vati ons,we can get  the  FI M  el e-  

I n t hi s  secti on.we wi l l  deduce  the CRLB f or TD est i m a-   t i on.In  thi s   ca s e,t he  DS  Wd  i s  s sum ed t a o be  a  known  param-  

ments  a s  f ol l ows.  

eter.W e  consi der  the  probl em  whe n t he  pha se   =

l s( …

 

) l 2 dt=2 SNR 

( 20)  

i s  unknown 

and known,whi ch are  0f ten  m et   i n  non-coher ent  and c ohe rent   recei vers,r es pe cti vel y. Note  t hat  part  of  the  FIM el em ent s  

2a2

s ( t一7 - ) l 。 dt=2SNR 

Re  

( 21)  

can be  reused and need not  be deduced repeatedl y.W e can 

de l e t e   t he   r e l a t i v e   c ol umns   a nd  r o ws   i n  Eq. ( 30)t o   g e t   t he   ne w  FI M.  

2a2

r t—r) l 。 dt=2SNI  ̄  

Re  

( 2 2)  

1. U nknow n phase  If   the  DS i s  known,t he  unknown vect or  wi l l   be  

2a2 

l s ( … oo  

 

2…

{ t ' 2+2 rt ' +r2   3)  

0=【 0 

T】  

( 3 4)  

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Cram er- Rao  Lower  Bounds  f or  Ti m e  Del ay  and  Doppl er  Shi f t  Esti m ati on 

637 

I n  thi s  case,t he unknown vector  wi l l  be 

a 

n 

0 

一 ∞pJ暑0  I u ∞0  0  Jo  0r Ju盘- I  > 

; g  

n  

一 一



一 ∞pJ盘0一 苗I u ∞ 0∞0 0  0兰 盘- I  > 

[ 0  

c a d] T 

( 40 )  

( 3   51  

1 

y 

Thus  

 

 

+ 

r   1 / 0 2  0 I( 8)=2 SNR  l   0  



m 

=  

c, )  

( 41 )  

L  0  

Ⅳ 0  R 

t 2j  

v ar ( &d ) CRLB2=( I/ 2SNR) ( I /<  ̄   )  

O  O   —

] I  

0 

1 

2  

2.K now n pha s e 

20 



40 



60 

( 42 )  

W hen  t he   pha se  i s   al s o  known,t he   unknown  vector   wi l l   be  =

[ a   d ] T 

( 43 )  

W e   can  obtai n 

0 

10 

20 

30 

40 

, = 2 S N R   罟 ]  

( 44 )  

va r ( &d) CaLB3=( 1 /2SNR) ( 1 / t 一 2 )  

( 45 )  

50 

SNR ( dB)  

V I.R el ati ons B etween C R LB s   Prom   the  above  res ul ts,we can  ind that f   2  一

1 =

1  =

 

 

口2  2一 Q2  

( 46)  

H ence 

va r (  ̄ ' ) CaL m  v ar ( #) CRLB2   v ar (  ̄ ' ) CRL B3  

( 4 7)  

W i th t he sam e  rea s on,we  can  al so  get  

SNR ( dB)   Fig. 1.Com parison of  the  vari ances  of  esti m ati on  obtai ned by 

si mul ati on  wi t h  the  CRLBs :(1)denotes   j oi nt   TD  and  DS est i mat i on;( 2)denot es  TD/DS es ti mati on wi th  unknown pha se;( 3)denotes  TD/DS es ti mati on wi t h  know n phase 

v ar ( &d ) CRLm  va r ( &d) CRL B2   v ar (  ̄d ) CRLB3  

( 4 8)  

I n f act,the  above  rel ati ons  ar e  not  occa si ona1 .I n s tati sti -  

cal   sense,knowi ng  the  i nf orm ati on  of   other  t hi ngs   can  reduce  the  uncer tai nt y of  i ts el f . Thi s  i s  a  wel l - known  theorem i n i n.  

f or m at i on  theory:condi ti oni ng  re duces  ent ropy[ 。, 引 So i n  our 

Hence,the  CRLB f or  TD i s 



probl em ,wi t h the decrea si ng of   the  num ber  of  unknown pa-  

va r(  ̄) CRL B2=( 1 /2SNR) ( 1 /  ̄ )  

( 36)  

r am eters,the es ti m at e vari ances  shoul d  decrea se  a s  the s am e   ti m e.  

Thi s   r e s ul t   i s   c ons i s t e n t   wi t h  t hat   i n  Ref . [ 9,PP. 72 ] .   2.K now n phase 

V II.Si m ul ati ons 

If   the  pha se  i s  al so  known,t he  un k nown  vector   wi l l  be 

0=[ 0  r] T 

( 3 7)  

In t hi s s ecti on,we  wi l l  use  M ont e Car l o si m ul ati on t o  dem onstr ate   the CRLBs.As  an  exam pl e,we  use 

Then 



2SN R 

]  

v ar (  ̄) CRLB3=f 1 / 2SNR) ( 1 /一 w2 )  

s ( t ) = ̄ /  ̄e x p{一  ( t —t c )  +歹  ( t —t   ) +j . l 。 ,、 t —t c )  }   ( 3 8)  

s  the s a ource  si gnal ,where  tc = 1,   c = 1,  

1. Usi ng 

. ( 1 5 ) 一( 1 9 ) ,we  ca n g e t   =t  ,t  = 1 /2+t :,0 =  ,   ( 39)   Eqs 22= 1 0 / 2+  +  /2   and  =  t  + / 2.The   CRLBs   c a n 

Thi s   r e s ul t   i s   c ons i s t e nt   wi t h  t ha t   i n  Re f . [ 8,PP. 2   1 4]   V .C R LB f or D S Esti m ati on 



be   e si a l y   obt ai ne d  ac c or di ng  t o  Eqs . ( 31 ) ,( 32) ,( 36) ,( 3 9) ,( 42 )   and( 45 ).Fo r   s i mpl i ci t y ,we   s e t   r= 0,0 2d= 0,a= 1 ,   0.   The m ea s urem ent  ti m e i s -5 

t  

5 and the s am pl i ng pe-  

r i od i s  0. 1.Because  the  M L est i m ator s  can attai n  the  CRLB 

As  i n previ ous  secti on,we  de duce  the CRLB f or  DS esti -  

s ym ptoti a cal l y,we  use   them  to  val i dat e  the  CRLBs  i n  our   si m-  

m ati on.I n our  de ducti on,we a s sum e  the TD  r = 0.I n f act,  

ul ati on.The   stati s ti cal  r esul ts  are  ba s ed  on 1 000  i ndepe ndent  

thi s  i s   not  necessi tous,but  i t   can  l ead to  si m pl er  expres si ons.  

tr i al s.The vari ances  and t he  CRLBs  are  shown i n  Fi g. 1.W e  

1.U nknow n pha s e 

can  see  that   the  vari ances   of   the  es ti m ati ons   are  very cl ose  t o 

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638  

Chi ne s e   Jo ur nal   o 1   El e ct r o ni c s  

200 5  

o r time del ay and  t he   CRLBs   at  hi gh  SNR.Thi s   s ho ws  t he  c or r e c t ne s s   of   o ur   [ 1 0】  B.Friedlander,“On the Cram er—Rao bound f

theor eti cal  deri vati on.  

Doppl er   es ti mati on .IEEE  ansact i ons   on I nfor m at i on The .   or y ,Vo1. 3O,No. 3,PP. 575—580,1984.  

[ 1 1 】  Q.Ji n,K. M.W ong,Z. Q.Luo,“ The  es ti mati on of   t i me del a y 

V III.Concl usi on 

and D oppl er stretch of wideband signal s”.IEEE 

nsactions 

on Signal  Processing,Vo1. 43,No. 4,PP. 904—916,1995.  

In thi s  paper,usi ng Fi sher  i nf or m at i on m at ri x,CRLBs  

f or  TD esti mat i on,DS esti mati on,and i oi nt  TD and DS esti —  

s ed approach  [ 1 2 】  X .X .Niu,P.C .Ching,Y .T.Chan, “Ⅵ velet ba

or f   j oi nt   ti me  del a y  and Doppl er   s tr et ch  mea s ur ement s  .I EEE 

mati on under  di ferent  condi t i ons ar e deri ved system ati cal l y.  

Transacti ons  on Aerospace and Electr o nic System s. Vo1 . 35.  

These  deri ved  bounds   ar e  consi st ent  wi th  the  pr evi ous   resul t s,  

No. 3.PP. 1111一l119.1999.  

and f urt herm ore.they can appl y to al l  SNRs. 、 ̄   al so di s—  

cuss   the   rel ati ons   between  the   CRLBs   obt ai ned  under   di ierent f  

[ 1 3 】A.J ak ob s s on,A. L.S wi n dl e hur s t ,P.S t o i c a,“ S ub s p a c e - ba se d   esti m ati on of ti m e del ays and D oppl er shi Ks”,IEEE Tr a nsac— 

ti ons on Signal  Processing,Vo1. 46,No. 9,PP. 2472—2483,1998.  

condi ti ons.I t  i s   shown  that  the  CRLBs  wi l l   decrease  wi t h  the  decrea si ng of  t he number  of   unknown parameter s.Fr om the  vi ewpoi nt   of   i nf o rmati on  theory.the  i nt ri nsi c  rea s on  or f   the   re—  

l at i ons   i s  f ur ther  gi ven.Si mul ati on  resul t s  are  pr esented that   show  a  good agr eement  wi th the  theoreti cal l y  deri ved ones.  

R ef erences  

【 1 】s. R.Doo l e y ,A. K.Nan di ,“ Adap t i v e  t i me   d e l a y   an d  Dop pl e r   shi ft  esti m ati on for narrow band signal s”. IEE Proceedings—   Radar,Sonar and  N avigation,Vl 01 .146,No. 5,PP. 243—250,1999.  

【 2 】Y.wu,H. c.s0,P. C.Chi n g,“ Jo i n t   t i me   de l a y   a nd   re f que n c y   esti m ati on via state—space real izati on”.IEEE Si gnat  Processi ng  Letters,Vl 0l _10,No.11.PP. 339-342.2003.  

[ 3 】A.  Pa p an dr e o u- su ppa p p0l a,  s. B.  Su ppa pp ol a,  “ S on a r  e c h o   ranging usi ng  si gnal s wi th nonli near  ti m e-f requency  characteri s— 

ti cs”,IEEE  Signal   Processing  Let ters,Vl 01.11,No. 3,PP. 393—396,   2004.  

[ 4 】w . Q.Zh an g,R.Ta o,“ Hi g h e r — or de r   wa ve l a nt  b a s e d  a pp r o a c h   f or   j oi nt   t i me  del ay  and  Doppl er   s tr et c h  es ti ma ti on”.Act a  El e c—   t o nica Sinica,Vo1 r . 33,No. 3,PP. 549 - 552,2005.  

[ 5 】Y. F.Ma,W . Q.Zha ng ,R.Tao ,“ Th e   c o mp l e x   a mbi g ui t y  f un c —   tion ba s ed on downsam pl ed f o urth-order stati sti cs”.  

i s currentl y a Professor.the Director of the Inf orm ati on Securi ty 

inese 

Research Center and the V i ce Chairm an  of  the Departm ent of   Elec.  

Jour nal   o f  El ec t r oni cs,vol _ 1 3,No. 4,pp. 701-704.2004.   『 6I  T. M .Cover ,J. A.Thoma s ,El ement s  of  Inf ormat i on Theor y,  

troni c Engi neeri ng. H e wa s  a vi si ting schol ar at  the Uni versi ty of 

John W i l ey & Sons,New York,U SA .1991.  

  1 7l  S. M .Kay,Fundam e nt al s  o f  St at i s t i cal  Si gnal  Pr oces si ng,  , 02 .   Est im at ion Theory,Prenti ce Hall,New Jersey,USA ,l993.  

[ 8 】T.Cha ng ,El e me n t s   o f   I n f o m at r i o n  Th e o y,Ts r i ng h ua   Uni v e r —  

Mi chi gan,Ann A rbor,U SA from  M ar. 2001 to A pr. 2002. Hi s 

research interests are in inf orm ati on securi ty.com m uni cati on & in—   o rm ati f on system .He ha s  publi shed m ore than 100  techni cal  papers  and three books. He is  a senior mem ber of the IEEE .He won the  Teachi ng and Research Award f or  O utstandi ng Young Teachers in 

si t y  Pr es s,Sei ji ng,Chi na,1993.  

Hi gher Educati on Instituti ons of  M OE,Chi na.in 2000 and C hi nese 

M . Y .Lin,Y . A .K e,Radar Signal  Theor y ,Nati onal  Defense In-  

O rdnance Sci ence and Technol ogy A ward f or  Young Speci ali sts in 

dus t r y  Pr es s,Bei ji ng,Chi na,1984.  

2003.( E—m ai l :r ant ao@ bi t . edu. cn1 .  

Cramer-Rao Lower Bounds for Time Delay and ...

ZHANG Weiqiang and TAO Ran. (Department of Electronic ... searches are not system atic and extensive. In this PaPer,. starting w ith a general m ...

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Jul 3, 2010 - metic operations in terms of the number of the given nodes in order to represent some ..... Taking into account that a generic n–.

Lower Bounds on Deterministic Schemes for the ...
of space and 2 probes is presented in a paper by Radhakrishnan, Raman and Rao[2]. ... BigTable. Google uses nondeterministic space efficient data structures ...

LOWER BOUNDS FOR RESONANCES OF INFINITE ...
D(z) of resonances at high energy i.e. when |Re(z)| → +∞. The second ...... few basic facts about arithmetic group. Instead of detailing the ..... An alternative way to contruct similar convex co-compact subgroups of. PSL2(Z) with δ close to 1 i

Square Deal: Lower Bounds and Improved Relaxations ...
Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery. Cun Mu, Bo Huang, John Wright and Donald Goldfarb. Introduction. Recovering a ...

Time Dispersion and Delay Spread Estimation for ...
of the proposed algorithms are compared against this theoretical limit. ..... The mobile speed used in the simulations3 is 60 km/h, and the time-varying channel is ..... sonal, Indoor Mobile Radio Commun., London, U.K., Sep. 2000, vol. 1, ... Air Int

Delay Spread and Time Dispersion Estimation for ...
where Xm(k) is the transmitted data symbol at the kth subcarrier of the mth OFDM symbol, and N is the number of subcarriers. After the addition of cyclic prefix ...

Bounds on Information Propagation Delay in ...
MAC contention, and introduce a random dynamic multi-digraph to model its connectivity. We first provide analytical results about the degree distribution of the ...

Lower bounds for a conjecture of Erd˝os and Turán
Jun 1, 2009 - n ∈ Nj. It follows that the union of the odd Nj satisfies. ⋃ j odd. Nj = {2a : a ∈ A}. In the language of their representation functions,. S1(z) + S3(z) ...

Lower bounds for a conjecture of Erd˝os and Turán
Jun 1, 2009 - Abstract. In this work we study representation functions of asymptotic ad- ditive bases and more general subsets of N (sets with few nonrepre-.

Non-trivial derandomization implies weak lower bounds
Aug 15, 2017 - fixed size and depth, one needs a non-trivial derandomization algorithm .... Journal of Computer and System Sciences, 65(4):672–694, 2002. 3 ...

Non-trivial derandomization implies weak lower bounds
Oct 17, 2017 - randomness r ∈ {0, 1}n , and a proof-oracle 7x ∈ {0, 1}2n for x, where n = n +. O(log(n)); the verifier issues poly(n) queries to 7x, verifies the ...

Lower Bounds on the Minimum Pseudo-Weight of ...
Nov 30, 2003 - indices are in Vr. We call C a (j, k)-regular code if the uniform column weight ..... Proof: In App. E of [14] the above lower bound for the minimum ...

Non-trivial derandomization implies weak lower bounds
Oct 25, 2017 - (Indeed, in this text we focus on derandomization of circuits with one-sided error.) ... The current text presents an alternative argument, in which. “weak lower .... Eli Ben-Sasson and Madhu Sudan. Simple PCPs with poly-log rate and

Message Lower Bounds via Efficient Network ...
Nov 28, 2016 - wireless sensor networks, where processors are powered by batteries with .... advantage of synchronicity in order to exploit silence, and uses ...

Minimax lower bounds via Neyman-Pearson lemma
May 7, 2013 - with Le Cam's inequality [3, Lemma 2.3]. Lemma 2. Let (S,S,µ) be a measure space, and let p, q be probability den- sity functions with respect to ...

TIME DELAY ESTIMATION: COMPRESSED SENSING ...
Sampling theorems for signals that lie in a union of subspaces have been receiving growing ..... and reconstructing signals of finite rate of innovation: Shannon.

RESONANCES AND DENSITY BOUNDS FOR CONVEX CO ...
Abstract. Let Γ be a convex co-compact subgroup of SL2(Z), and let Γ(q) be the sequence of ”congruence” subgroups of Γ. Let. Rq ⊂ C be the resonances of the ...

Delay learning and polychronization for reservoir computing
Feb 1, 2008 - At any time (initialization, learning and generalization phases), the complete cartography of the network activity can be observed on a spike ...

A new method to obtain lower bounds for polynomial ...
Dec 10, 1999 - Symposium on Applied Algebra, Algebraic Algorithms and Error Correcting. Codes, AAECC-5, Lecture Notes in Computer Science vol.

Simultaneous Technology Mapping and Placement for Delay ...
The algorithm employs a dynamic programming (DP) technique and runs .... network or the technology decomposed circuit or the mapped netlist is a DAG G(V, ...

Two-Stage Method for Joint Time Delay and Doppler ...
the pre-weighted Zoom FFT method is used for fast computing the ambiguity function and ..... biguity function in the discrete grid points with relatively large steps.

Processing Time Bounds of Schedule-Preserving DEVS
This paper proposes a class of discrete event system specification (DEVS), called schedule-preserving DEVS (SP-DEVS), .... Let's consider a controller for a crosswalk light system shown in Figure. 2(a). In the system, there are two traffic ...... In