(angle modulation) u(t) = Umcos(wt+j) Um , s(t) w, j. , u(t) y(t) = wt+j, .
(, phase modulation - PM). wo s(t). , :
u(t) = Um cos[wot + k×s(t)], (9.2.1)
k . . 9.2.1.
. 9.2.1. . |
s(t) = 0, uo(t). s(t) y(t)=wot+k×s(t) wot. , s(t) . s(t) Dy wot ( Dj = k×smax(t), Dj = k×smin(t) ).
(instantaneous frequency), :
ω(t) = y(t)/dt = ωo + k ds(t)/dt.
:
y(t) = ω(t) dt, y(t) = ω(t) dt +jo.
(, frequency modulation - FM) , wo :
w(t) = wo + k×s(t). (9.2.2)
, :
y(t) = ωo(t) + k s(t) dt, y(t) = ωo(t) + k s(t) dt +jo.
:
u(t) = Um cos(ωot+k s(t) dt +jo). (9.2.3)
, Dw = k×smax(t), Dw = k×smin(t).
. , , . (). , , , s(t) .
. ω. :
j(t) = b sin(Wt),
b - (modulation index), . ω:
y(t) = wot + b sin(Wt).
:
u(t) = Um cos(wot + b sin(Wt)). (9.2.4)
:
ω(t) = dy(t)/dt = wo + bW cos(Wt).
, , . ω ωd = bW, (frequency deviation). , :
b = ωd/W. (9.2.5)
W .
W, :
b = const, ωd = b W.
, , :
ωd = const, b = ωd/W.
.
(9.2.4) :
u(t) = Umcos(b×sin(Wt)) cos(wot) - Umsin(b×sin(Wt)) sin(wot). (9.2.6)
(b<<1, ) :
cos(b×sin(Wt)) 1, sin(b×sin(Wt)) b×sin(wot).
(9.2.6), :
u(t) Umcos(wot) + (bUm/2)cos[(wo+W)t] + (-bUm/2)cos[(wo-W)t]. (9.2.7)
(9.1.4) , b<<1 wo+W wo-W. , .. 1800 . , 180 . , b .
. 9.2.2. . |
b (9.2.4) :
u(t)=Um Jk(m) cos[(wo+kW)t],
Jk(m) k- m=b. , - , wokW, , , Jk(m). Um=1 . 9.2.2.
b . b , . , ω . . 9.2.2 , (2.405, 5.52, 8.654 ..) wo . . 9.2.3.
, , . :
= 2(b+1)W, (9.2.8)
.. k>(b+1) . , , b>>1, :
2bW = 2wd. (9.2.9)
. 9.2.3. .
( 2500 , 25 , )
, , 2W, , b . , .
: J-k(m) = (-1)kJk(m). , wo+kW wo-kW k, 180 k.
. , woW1W2...Wi, Wi. .
.
:
ua(t) = u(t) + j uh(t),
uh(t) u(t), u(t) (1/πt):
uh(t) = (1/π) u(t') dt'/(t-t').
:
y(t) = arg(ua(t)).
. ωt:
j(t) = y(t) - ωot.
ω:
j(t) = y(t)/dt - ωo.
, , , .
, 90:
u1(t) = u(t) cos(ωot) = Um cos(ωot+j(t) cos(ωot) = ½ Um cos j(t) + ½ cos(2wot+j(t)),
u2(t) = u(t) sin(ωot) = Um cos(ωot+j(t) sin(ωot) = - ½ Um sin j(t) + ½ sin(2wot+j(t)).
, :
ua(t) = ½ Um cos j(t) - ½j Um sin j(t).
, , , .
, . - :
s(t) = u(t) cos(ωot+j(t)).
s(t) , . -.
s(t) = u(t) cos(ωot) cos j(t) u(t) sin(ωot) sin j(t).
a(t) = u(t) cos j(t) b(t) = -u(t) sin j(t), a(t) b(t) cos(ωot) sin(ωot), 90 :
s(t) = a(t) cos(ωot) + b(t) sin(ωot).
(quadrature), - ().
(9.1.17) :
S(ω) = ½ A(ω+ωo) + ½ A(ω-ωo) ½j B(ω+ωo) + ½j B(ω-ωo).
, 90:
s1(t) = s(t) cos ωot = ½ a(t) + ½ a(t) cos 2ωot + ½ b(t) sin 2ωot,
s2(t) = s(t) sin ωot = ½ b(t) + ½ a(t) sin 2ωot - ½ b(t) cos 2ωot.
a(t) b(t) . , .
Mathcad.
.
N:= 2999 n:= 0.. N Dt:= 0.001 ' ( ).
f0:= 50 f1:= 2 f2:= 3 ' , .
s1n:= sin(2pf1nDt) ' ( 1).
s2n:= sin(2pf2nDt) ' ( 1).
b:=10 jn:= bs2n ' s2n
un:= s1ncos(2pf0nDt+jn) '-
U:= CFFT(u) Df:= 1/[(N+1)Dt] '
an:= s1ncos(jn) bn:= s1nsin(jn) '
sn:= ancos(2pf0nDt) + bnsin(2pf0nDt) ' .
'un ,
', , .
.
u1n:= sncos(2pf0nDt) ' an bn.
u2n:= snsin(2pf0nDt) ' u2n bn -2 .
U1:= CFFT(u1) U2:= CFFT(u2) ' , .
M:= 50/Df m:= M.. N+1-M U1m:= 0 U2m:= 0 ' ( 50 ).
u3:= ICFFT(U1) u4:= ICFFT(U2) ' .
' u3n u4n 2
' c an bn.