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(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.





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