Ch3_ans

|n2 − n1 | 2n1 n2
=
n1 ,n2 →∞
lim
1−
1 1 − 2n1 2n 2
=1
¤±ÅL§äkþëY5" þ5µ
τ →0
lim RX (τ ) = lim
τ →0
sin ατ = α = RX (0) τ
4
éR
X (t1 , t2 )
¦2Âê
∂ 2 RX (t1 , t2 ) ∂t1 ∂t2 t1 =t2 =t 1 = lim [RX (t + τ1 , t + τ2 ) − RX (t + τ1 , t) − RX (t, t + τ2 ) + RX (t, t)] τ1 ,τ2 →0 τ1 τ2 sin α(τ1 − τ2 ) sin ατ1 sin ατ2 1 − − +α = lim τ1 ,τ2 →0 τ1 τ2 τ1 − τ2 τ1 τ2 = α(τ1 − τ2 ) − 1 τ1 ,τ2 →0 τ1 τ2 lim − = lim ατ1 −
z
z
fY (y )dy + fY (z )
−∞ −∞
fX (x)dx
ÅCþZ 'A¼ê
ΦZ (ω ) = E {ejωZ } =
∞ −∞
∞ −∞
ejω y fXY (x, y )dxdy
x
2
-z = x/y§Kdz = dx/y§¤±
0 −∞ ∞ ∞
ΦZ (ω ) =
−∞ ∞
dy
∞ ∞
ejωz fXY (zy, y )y dz +
eΘ(−π, π)þþ!©Ù§ÁéÅL§X (t) = f (t + Θ)?1Fourier?ê Ðm" )µ éf (t + θ)?1Fourier?êÐm
∞
f (t + θ) =
n=−∞
cn ejnt ,
t ∈ (−π, π )
5
Ù¥§
cn = ====== 1 = 2π
π
-
1 2π
π
f (t + θ)e−jnt dt
2 1 fX (x) = √ e−x /2 2π
ù§¨y > 0§
fY (y ) = fX (x1 ) dx2 dx1 + fX (x2 ) dy dy 1 1 √ √ = fX ( y ) √ + fX (− y ) √ 2 y 2 y 1 =√ e−y/2 2πy
¨y < 0§w,f nÜå5§
−π
x=t+θ
1 2π
π +θ
f (x)e−jnx · ejnθ dx
−π +θ
f (x)e−jnx dx · ejnθ
−π
£Ïf (x)Úe '±ÏÑ´2π¤
−jnx
= an ejnθ
¤±X (t) = f (t + θ)'Fourier?êÐm
∞ ∞
f (t + θ) =
n=−∞
an ejnθ ejnt =
fZ (z ) = FZ (z ) = fX (z )
z
fY (y )dy + fY (z )
z
fX (x)dx
z } = P {max(X, Y )
z
z } = P {X
z
z, Y
z}
z }P {Y
z} =
−∞
fX (x)dx
−∞
fY (y )dy
¤±
fZ (z ) = FZ (z ) = fX (z )
Y
(y ) = 0
"
e−y/2 fY (y ) = √ U (y ) 2πy
1
X ÚY ´üpÕáÅCþ§ ÙVÇݼê©Of (x)Úf (y)§ Á¦eÅCþVÇÝ¼êµ Z = X + Y ¶ Z = min(X, Y )¶ Z = max(X, Y )¶ Z = X/Y ¶ Z = XY " )µ ÏX Y pÕ᧤±f (x, y) = f (x)f (y)"
0
er
2
/2
dr
er
0
/2
dr
l k
fW (w) = FW (w) =
2 w 2 /2 , e π
w
0
3-19
®mSX g'¼ê
n
RX [n1 , n2 ] = 1 −
Áy²X þÂñ" y²µ Ï
n n1 ,n2 →∞
|n1 − n2 | 2n1 n2
lim
RX [n1 , n2 ] =
n=−∞
an ejn(t+θ) ,
t ∈ (−π, π )
ˇ (t)X (t))ÛL§" X (t)¢L§§¡ÅL§Z (t) = X (t) + jX Á y²)ÛL§k±e5µeX (t)¢°²­L§§KÙ)ÛL§Z (t)½ ˇ (τ )]" °²­L§§ R (τ ) = 2[R (τ ) + jR )µ ˇ (t)°²­"¿ ÏX (t)°²­§¤±X 3-39
âþOK§X (t)þ" þÈ5µ
∞ −∞ ∞ ∞ −∞ ∞
RX (t1 , t2 )dt1 dt2 =
−∞ ∞
=
−∞ ∞
sin α(t1 − t2 ) dt1 dt2 t 1 − t2 −∞ ∞ sin α(t1 − t2 ) dt1 dt2 t1 − t2 −∞
=
−∞
π dt2
∞ π
=A = A 2π
fΨ (ψ )dψ
−∞ ∞
fΦ (φ) sin(2πψt + φ)dφ
−π π
fΨ (ψ )dψ
−∞ −π
sin(2πψt + φ)dφ
=0 X (t)
'g'¼ê
= A2 E {sin[2πΨ (t + τ ) + Φ] sin(2πΨ t + Φ)} 1 = − A2 E {cos(4πΨ t + 2πΨ τ + 2Φ) − cos(2πΨ τ )} 2 1 1 = − A2 E {cos(4πΨ t + 2πΨ τ + 2Φ)} + A2 E {cos(2πΨ τ )} 2 2 1 2 ∞ =0+ A fΨ (ψ ) cos(2πψτ )dψ 2 −∞
g'¼ê
RZ (t + τ, t) = E {Z (t + τ )Z ∗ (t)} ˇ (t + τ )][X (t) − jX ˇ (t)]} = E {[X (t + τ ) + jX ˇ (t + τ )X ˇ (t)] + j[X ˇ (t + τ )X (t) − X (t + τ )X ˇ (t)]} = E {[X (t + τ )X (t) + X = RX (τ ) + RX ˇ (τ ) + j[RXX ˇ (τ ) − RX X ˇ (τ )] ˇ X (τ ) + R ˇ X (τ )] = RX (τ ) + RX (τ ) + j[R ˇ X (τ ) = 2RX (τ ) + 2jR

FZ (z ) = P {Z =1−
z } = 1 − P {Z
∞ ∞
z } = 1 − P {max(X, Y ) > z }
= 1 − P {X > z, Y > z } = 1 − P {X > z }P {Y > z } fX (x)dx
z z ∞ ∞
fY (y )dy
¤±
FZ (z ) = P {Z = P {X
0
dy
−∞
ejωz fXY (zy, y )y dz
=
−∞ ∞ −∞
ejωz fXY (zy, y )|y |dz dy
∞
=
−∞ ∞
ejωz
−∞ ∞
fXY (zy, y )|y |dy dz fX (zy )fY (y )|y |dy dz
−∞
=
−∞
ejωz
ÏdÅCþZ 'VÇݼê
∞
fZ (z ) =
−∞
z ,y y
1 dz y
ÏdÅCþZ 'VÇݼê
∞
fZ (z ) =
fX
−∞
z y
fY (y )
1 dy y 1 dx x
þª±
fZ (z ) =
∞
fX (x)fY
−∞
z x
3-11
X, Y,√ Z pÕá! "þ!ü GaussÅCþ§Á¦ ÅCþW = X + Y + Z VÇݼê" )µ
Z X X
ˇ (t)} = 0 E {X RX (τ ) = RX ˇ (τ ) ˇ X (τ ) RXX ˇ (τ ) = −RX X ˇ (τ ) = R
)ÛL§Z (t)'þ¼ê
ˇ (t)} = E {X (t)} + jE {X ˇ (t)} = mX mZ (t) = E {X (t) + jX
∞
RX (t + τ, t) = E {X (t + τ )X (t)}
= A2 X (t)
fΨ (ψ ) cos(2πψτ )dψ
0
'¼ê
var{X (t)} = CX (t, t) = RX (t, t) − m2 X (t) = RX (t, t)
5ÅL§61nÙY
3-1
X ´IOÅCþ§Á¦Y = X VÇݼêf (y)" {µ ÅCþY 'A¼ê
2 Y ∞
ΦY (ω ) =
−∞
ejωx fX (x)dx
∞ 0
2
-y = x §K
2
2 =√ 2π √ dy = 2xdx = 2 y dx 2 ΦY (ω ) = √ 2π
Hale Waihona Puke Baidu
6
ÅL§X (t) = A sin(2πΨ t + Φ)§Ù¥A~ê§Ψ ÚΦpÕá ÅCþ§Ψ VÇݼêó¼ê§Φ(−π, π)Sþ!©Ù§Áy²µ X (t)°²­L§¶ X (t)´þH{" )µ X (t)'þ¼ê
3-41 E {X (t)} = E {A sin(2πΨ t + Φ)}
fX (zy )fY (y )|y |dy
−∞
þª±
fZ (z ) =
∞
fX (x)fY
−∞
ÅCþZ 'A¼ê -z = xy§Kdz = ydx§¤±
0 −∞
x z
|x| dx z2
∞
∞
ΦZ (ω ) = E {ejωZ } =
−∞ −∞
ejωxy fXY (x, y )dxdy
ΦZ (ω ) =
2 2 2
3
ÏX, Y, Z pÕá'!"þ'!ü 'GaussÅCþ§¤±
2 2 2 1 fXY Z (x, y, z ) = fX (x)fY (y )fZ (z ) = √ e(x +y +z )/2 3 ( 2π )
¨w < 0§w,kf
W (w )
=0
¶¨w 0k
e
w2
=∞
¤±X (t)3(−∞, +∞)SØ´þÈ'"¢´e½k«m§~ X[a, b]§Ù¥a, b ∈ R§KX (t)´þÈ'" 3-34 ®f (t)Rþ±Ï2π ±Ï¼ê§ f (t)3(−π, π )þkXeFourier? êÐm
∞
f (t) =
n=−∞
an ejnt
∞
ejωx e−x
2
2
/2
dx
§¤±
∞ 0
dy ejωy e−y/2 √ 2 y
=
Ïd {µ ⮧
2
e−y/2 U (y ) dy ejωy √ 2πy −∞
e−y/2 fY (y ) = √ U (y ) 2πy
©¼êy = g(x) = x "éu?¿'y > 0§3
x1 = √ y, √ x2 = − y
1 3 3 3! α τ1 1 3 3! α (τ1
− τ2 )3 + o(τ1 − τ2 )3 τ1 − τ2 − ατ2 −
1 3 3 3! α τ2 3 + o(τ2 )
3 + o(τ1 )
τ1
τ2
+α
1 α3 2 1 2 2 τ1 + τ2 − (τ1 − τ2 ) = α3 < ∞ τ1 ,τ2 →0 τ1 τ2 6 3
−∞ ∞
dy
∞ ∞
ejωz fXY ejωz fXY
=
−∞ ∞ −∞
∞
=
−∞ ∞
ejωz
−∞ ∞
fXY fX
−∞
=
−∞
ejωz
∞ 1 z ,y dz + dy y y 0 1 z ,y dz dy y y 1 z ,y dy dz y y z 1 fY (y ) dy dz y y
∞
ejωz fXY
3-4
X Y XY X Y
FZ (z ) = P {Z
∞
z } = P {X + Y
z −x
z} =
x+y z
fXY (x, y )dxdy
=
fXY (x, y )dy dx
−∞ −∞
¤±
fZ (z ) = =
dFZ (z ) = dz
∞ −∞
∞
fXY (x, z − x)dx
−∞
fX (x)fY (z − x)dx = fX (z ) ∗ fY (z )
x2 +y 2 +z 2 2
FW (w) =
1 (2π )3/2
x2 +y 2 +z 2
dxdy dz
È©Cx = r sin ϕ sin θ, y = r sin ϕ cos θ, z = r cos ϕ§K
FW (w) = = 1 (2π )3/2 2 π
w 2π π w
dθ
0 0
2
sin ϕdϕ
n1 ,n2 →∞
lim
1−
âLo` eveOK§ X þÂñ" 3-28 ?ØeÅL§þëY5! þ5ÚþÈ5µ X (t)´"þ g'¼êR (τ ) = sin ατ /τ °²­L§§Ù¥α > 0" )µ þëY5µ @R (0) = α"ù§
n X X