@,s1 lect #258;b #4-21-99


  ,suppose t ! *annel is
frequ5cy selective1 al? /
sl[ly v>y+4  ,t is1 we /
assume ,t "k""k ,t;coh, b t
w .1 ,b;c_4  ,if ;v(;t) is !
,,lp equival5t -pil] 5velope
(! transmitt$ signal1 !n ;v(
;t) is b& limit$ to ?w/2#_4
,9vok+ ! sampl+ !orem1 we h
  u(t) .k ".,s%n .k -,=<,=]u(
    ?n/w#),?sin #2.p?w/2#(t-?n
    /w#),/2.p?w/2#(t-?n/w#),#
  u(t)"$[33o<,f],u(w) .k ?1
    /,w#".,s%n .k -,=<,=],u(?n
    /w#)e^-?jnw/w#",p.pw^(w)
    t]m 0t]m ,f\ri] trans=m
  ": ,p;a"(x) .k .(

  #1,   \x\ "k: a
  #0,   else

                            #a
    lect #258;b #4-21-99    #a

  ?sin w;c"t/.pt#
    $[33o ,p;w;;c"(w)
  ,figure3  #2;,d graph4
;x-axis label$ ;x1 ;y-axis
label$ ,p;n"(x), a /ep graph )
magnitude #1 f -a 6a on !
;x-axis1 z]o "ey": else4

  ,!n ! lp equiv4 \tput (!
*annel is giv5 by3
  x;l"(t)
    .k ?1/2.p#!;-,=^,=
      "c(f_2;t),u(w)e^jwt
      "dw   w .k #2.pf
    .k ?1/w#".,s%n .k -,=<,=]
      u(?n/w#)!;-,=^,=
      "c(f_2;t)e^jw(t-?n/w#)
      "dw,p;.pw"(w)dw
  ,f ,proakis3
  x;l"(t)
    .k ?1/w#".,s%n .k ,=<-,=]
      u(?n/w#)c1(t-?n/w#jt)
                            #b
    lect #258;b #4-21-99    #a

      n correct un.s ,,bw at
      c(f_2;t) .k ,,bw (
      ,p;.pw"(w)
    or if we def9e c1(.t_2;t)
      6be 9v4 fit at
      ,c(f_2;t),p;.pw"(w)
    c1(t-?n/w#jt)"$[33o<,f]
      c(f, t)  n correct
  ,s9ce x;l"(t) is a 3volu;n1
we c express x z
  x;l"(t) .k ?1
    /w#".,s%n .k -,=<,=]u(t
    -?n/w#)c1(?n/w#_2;t)
  ,if we def9e c;n"(t)
".k<$t] ?1/w#c1(?n/w#_2;t) z !
n? *annel coeff1 we h
  x;l"(t)
    .k ".,s%n .k -,=<,=]c;n
    "(t)u(t-?n/w#)
  ,?us1 ! ,,lp equiv4 impulse
resp41 &! ,,lp equiv4 transf]
func;n >e giv5 by
                            #c
    lect #258;b #4-21-99    #a

  c1(.t_2;t) .k
    ".,s%n .k -,=<,=]c;n
    "(t).d(.t-?n/w#)   impulse
    response ( tapp$-delay l9e
  c1(f_2;t) .k
    ".,s%n .k -,=<,=]c;n"(t)e^
    -j?.wn/w#
  ,fur!r1 s9ce ! l9e span (!
multipass is ,t;m, we c
truncate ! abv sums4  ,let1
  x;l"(.g) .k ".,s%n .k #1<,l]
    c;n"(t)u(t-?n/w#)
  ": ,l ".k<$t] @(,?,t;m"+?1
    /w#,/?1/w#,#@)
    .k @(,t;m",w@)+1   @)
    denotes 9teg] "p
  & ": ?1/w# 9 ! num]ator
    >ises f ! "t spr1d+ 9h]5t
    9 ! *annel repres5t,n
  c1(.t_2;t)
    $[33o c(f_2;t),p;.pw
    "(w)   ,p;.pw"(w) $[33
                            #d
    lect #258;b #4-21-99    #a

    pulse fan a4s approx4 ?1
    /w# ( wid? to c(.t_2;t)
  ,a tapp$ delay "t model = ?
*annel is %[n 2l

  ,figure3  block diagram

  ,notice t ea* (! "c%:];i"(t)
>e -plex z]o m1n ,gaussian r&
processes1 & t 2c (!
uncorrelat$ scatt]+ assump;n 9
! ,,wssus (,,us.kuncorr4
scatt) model1 !y >e all
/ati/ic,y 9dep5d5t4

  ,if we n[ 3sid] pass+ a
,,bpsk signal "? ! *annel1 &
assume t ,t .1".1 ,t;m (ie c
ignore ,,isi) !n ! e6ective
,,isi c 2 ignor$4  ,! te/
/ati/ic (! optimum rcvr is
giv5 0! foll[+3
                            #e
    lect #258;b #4-21-99    #a

  "v%:](,t) .k ,re .(!;0^,t
    ""r%:];l"(t)
    .,s;n ;.k #1^,l"c;h^@#
    "(t)u^@#"(t-?k/w#)dt.)
    correlator (,,mf at
    correct sampl+ "t)
  ,note3"r%:] $[33 rcvd wave=m
    (lp equiv)
  ,note3".,s%n .k #1<,l]c;h^@#
    "(t)u^@#"(t-?k/w#) $[33
    tapp$ delay l9e z \r rcvr
    /ructure
  ": "r%:];l"(t) is ! -plex
5velope (! rcvd 5velope
"r%:](t)_4  ,note t !
imple;t,n ( ? rcvr is a tapp$
delay l9e & is "kn z a ,,rake
rcvr4

  ,figure3  block diagram4
,note #1_3  3j4 /rips (f
phase4  ,n[ we ne$ c>ri]
                            #f
    lect #258;b #4-21-99    #a

track+4
  ,note #2_3  we re,y ne$
sep>ate 9tegr,ns = ea* tap4 (
ov] xs [n ?j/w#)

  ,figure3 block diagram4
,note3  ,? design aligns all
multipa? seg;ts s we don't ne$
sep>ate 9teg] t]ms = ea* tap4

  ,figure3  ,block diagram4
,c>ri] track+4

  ,assum+ t ! v>i,n 9 ! coeff4
is su6ici5tly sl[ s t ! .(c;h
"(t).) c 2 le>n$1 & 1 9
"picul>1 t !y >e 3/ant )9 any
bit 9t]val4
  ",v%:](,t) .k ,re .(
    .,s;k ;.k #1^,l""c%:];h^@#
    "!;0^,t""r%:];l"(t)u^@#"(t
    -?k/w#)dt.)
                            #g
    lect #258;b #4-21-99    #a

  .k ,re .(.,s;k ;.k #1^,l
    ""c%:];h^@#"!;0^,t"@(
    .,s;n ;.k #1^,l""c%:]
    ;n",u(t-?n/w#)+"n%:];l
    "(t)@)u^@#"(t-?k/w#)dt.)
  ": "r%:];l"(t) $o @(
    .,s;n ;.k #1^,l""c%:]
    ;n",u(t-?n/w#)+"n%:];l
    "(t)@)
  ,assume ! wav=m v(t) is
design$ s* t
  !;0^,t"u(t-?n/w#)u^@#"(t-?k
    /w#)dt @:.k #0   = n /.k k
  !n ",v%:](,t) .k ,re .(
    .,s;h ;.k #1^,l"\c;k^2"\
    !;0^,t"u(t-?h/w#)u^@#"(t-
    ?k/w#)dt+.,s;k ;.k #1^,l
    "c;h^@#"!;0^,t""n%:];l
    "(t)u^@#"(t-?k/w#)dt.)
  @:.k ,re .(2,e
    .,s;k ;.k #1^,l"".a%:];k^2
    "+.,s;k ;.k #1^,l"".a%:];k
                            #h
    lect #258;b #4-21-99    #a

    ""n%:];k".)   ": "n%:];k
    ".k<$t] e^-j.?^;k"!;0^,t
    ""n%:];l"(t)u(t-?k/w#)dt
  ,approx4 s9ce = ea* %ift
    $o ?k/w#, we lose "s 5]gy
    \tside 9tegral !;o^,t_4
    ie assume ?,l/w# "k""k ,t
  ,note t ",v%:](,t) (te/
/ati/ic) is id5tical 6! te/
/ati/ic t we f.d =
manual-ratio-comb9+ ) ,l? ord]
div]s;y4  ,?1 = p]fect *annel
e/imates1 ! func;n rcvrs >e
equival5t4

  ,figure3  simple block
diagram
  a(t)cos .w0t+b(t)sin .w0t
    .k ,re .(u(t)e^j.w^;0^t
    ".) $o ,b,p,f $o y(t)
    .k ,re .(x;l"(t)e^-j.wt".)

                            #i