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In
so
Lets
BPSK
say
(t)
+
random
is
our
x
fo
=
cos(29 fot + 5 r)
symbol
is
2
.
3
of
process
each
space
=
symbols
corresponds to
·
in
·
out
[000 007 010,
#
,
,
.
1 bit.
phase - 'O'
phase ...,
717]
'
W
100 N
·
&
O
&
&
&
False
Lets
.
X(H) round(0)
=
When
+20
Otherwise
say & is
+
t
.
X0)
XE) is
a
Garsion
.
R V
discrete
R V
.
.
O
is
a
continuous
.
When
they
RV/RP
a
RV
we
this
because
#
(x)
=
ask
which
to
need
is
=
to
calculate it
continous
a
0(X([x)
dependent
is
Fy(x)
=
e-
!
.
P(t w = x)
-
P(wSt x)
-
=
1
-
-w)
e
S
-
t
-
-
Fw(t x)
-
+ y
endw =
(t x)
CDF
R V
**
=
another
a
from
=
Fr(t -x)
x>0
↑
Because
of
PDF ordCDF
a
us
w0
-
(x
=
1
-
e
-
t)
Fy(x)
=
After
n
writing
(t -+)
Tor
P(X()(x)
=
P(e
CDF
the
complicates
the
E
-
+
u(+ T) < x)
-
.
need to notice
we
CDF
UC-T) is
.
either
0
.
5/t
When
P(u(t
-
)
=
0) = P( + jt)
1
=
-
( 1) = P(T(H)
=
=
=
P(X(t) )
=
Because elt-0
When Tht
P(u)
F(H)
F(t)
=
P(x()
-
=
(
e
-
T)
P(t((nx + tad +< t)
+ ((nxad+(t)
P(T
(t)
↓
p(e
0(4
F ((y + +
+ T)
-
(xad+
=
+
=
=
-
)
(ex + /x
+=
x -
P( + ((ny + t)
Now
↑ (xH)
dirac
both
parts
0) = T F((t)
means
we
=
write
-
at
'O'
in
the
fy(xi ( F ( )) f(x)
=
-
+
+
.
+
that there
PDF of
X (t)
f + (kx
t)/x
+
is
a
Puppy
Because
fails
bernoulli
between
independent.
trials
different
are
i it
is
independent
connective
# of
successes are
When
leave
2 minute
exactly
After
(X + e
-
fN()
facel
enters
is
still
bank.
in the
more
arrivial .
after
249 minute Whoever
So until the
he will
bank
the
arrives
customer
a
=
.
2 minutes
2 minutes
=
12x1
"
+n!
the
only
,
than
"
ago
,
are
**!
e
...
Oft < L
people
n=
who
arrived no
still inside .
D ...
t7, 2
&
Probability
n= 0
,1
of
'n'
people
arrive
in
2 minute
.
We
first
divide
the
A , (H) :
# of
7 minute
customers
still
at service
4
Il
k
customers
A2()"2
Since
can
first
customers
find
at back
service
at
Same question
need to
adjust
the
fai It"
=
:
minute
time
't :
previous
parameters
-
e
are
of 1
the
as
xt
In
=
W
equiprobable we
are
rates
arrivial
their
say
Lets
these
in 2 groups.
/2
customers
one
,
+ 77
(E.1)". ent 1
/
we
in
just
*
= (t)c Y
*
e
/
(2
+x
+7 2
,
.
.
T
TPPof
His
&
Ac(H Az()
+
AnCH)
of
And
fa
=
(1)"
e
AzH)
that
add two Poisson
when
we
arrivial
Process
e
·
know
we
we
just
arrivial rates-
the
Sum
(x +.
and
+ <
Yn
**
:
(x)" e(z)/n:
1 +2
+
72
.
Blo
·
·
=
Brownian
for
Conditions
0
B( + + z) BH
-
X(0)
=
0
=
Given
=
.
-
+
+) y(x)
-
process
N(o)
Or
y10 = X ( %
X(ct + c) X(ch
y( +
motion
=
=
X(ct
N(0 M) Given
+
(2)
-
X(ct) =NE
V
9)
full
X() =
e-"who
:
else
⑧
E(t-w]
5)
(x(
.
2)
-
E(w]
+-
=
E(X( X(t T)] My(y) Mx(x
+
-
-
-
+
+
Extw +* zw-ew]-Y (2
=
-2
-
+2+ +
2 + -F
1
O
TE[m]
ELw]]-2 ECW]
=
+
O
2
=
=
t
=
1
2t
-
+
-
z
-
+
Fly
=
P(Y) < y(a) =P(aX(y)
↑
=
Ec
ELX H)]
&
9) Any
power
=
<
6) Elcos(2nfct ]
+
means
Ycoskf
totite
=
,
(sin12fct +
+2
=
-
=
1
do
sin
(efet)
O
=
2) Y()
=
X (t)
E(Y()]
=
cos(24 fet of
+
E(X(1] E[cos (2mfc o 0
++
↑
X
Because
=
and
O are
independent
dE[Y4H] =?
=
=
Ef
1
.
X H) 105122fc ++ ol] ELX HD EL coil
=
.
.
ELE (1
.
+
] E
es(2 (efc +
.
=
E((os) 2 (2Mfc + +)] 0
.
+
=
9) ECXc()]
Rx (i)
=
,
ECX(t + ]
=
=
My
Because X is
WSS
E(X (t) X (t z)] E[X(t a) x(t d))
-
+
=
+
-
.
+
.
T+
E(X(t) x(t + T] Ry(z)
=
.
=
Because
Rxx (T) E(X(t) x(t + i a)]
=
We
so
b)
+
.
,
that
see
they
cross
=
Rxx(t)
+
=
M=
-
E(X(a(t T)) X(a +)]
+
=
independent from 't
is
jointly Was
are
=
Ry(z a)
correlation
E(XcH] E[X(at)]
Rx (i)
=
X is WSS
E(X (at)
=
.
·
x(t +
My(az)
z)) Rx((
=
-
1) +
-
2)X
not crosWSS
-
ECTCH]
=
ELNul daJe ECNI : Men
.
t Ma
=
.
Myk E(Y() Y(t
=
.
+
z)]zE[Nml
.
N(r)
.
dadr]
=** E(NH N(vl]dadv
.
ooh
+
t +7
=SS
Rix
a.
f(u-r] dud
.
Notice
only
of the
peak
the
if
integral
is
inside
within
integral
will
to
result
'O'to 't + t. So the
overlaping
boundries
'7
result of
length of 'O' to't and 'O toty
t+ 2
t
(S
0
is
'GG)' is within the
of the
Therefore
.
' v'
only when
double integral
S(t) is equals to 17
integral of
that
a
.
S(u-v) dudu
=
min(t
t+
,
2) .
o
Because TH)
process.
is
not
WSS it
,
is
constationary
a
RylinE( Yel Y(t z)] E(X(nt XIdt mul
.
Rx(2
R (nt)
+
=
.
En Self
,
+
&
Sx()
=
R(ht
Forfouriear ele
Sy(f) (H( + )(S ()
=
+
=
*
H (f) H (f)
.
Sx (f)
(ch Thefterft) (det5ft) .
=
=
.
.
I
an + an
E Catal Rolei
·
did (e 22 fltztal+
+
+
and
ef(t)
+
Rx-tettel Rx(2-tutt)
+
.
Sett
a)
Sxy(f) F[Rx(z)) F(hell
=
·
= +p
-
7 The f
= 8
.
1
.
4-524f
u
From fourier table
b)
Rxy(z) = F "(Sxe(fi)
SxyHi First
C
1
.
4 the
+
L
7+ 529f
a=
)
+ 24f =
=
8
.
%
+ 24f =
b =)
8
-
.
a
=
52Mf = - 7
8
4
J 5
.
=)
4
.
=
G
=
b
l
Sxy( =
e +
g&
4 +29f
-
7 +1297
4 5297
+
F
S
⑳
&
O
4+ J
(
2rf
Jet
71
Rxy(t)
=
11:
=
41
42
% -eu(t
-
-