"!#$%#&(' )+*-,"./10#2 53 4768479:4A@:B/C)@:476D4E9:47F:GHIFKJLFNMPO/JLHQ4 RTS 6VU)OW@:XZY)[8;\HKXZ]KY)HI]7C;\XZ6 S Y_^9`47F`]I9baY)Xdce]K9NF`Xd@">
!"$#&%'%)(*+,%-$..0/213-154246+7.0+,8$%:9.&/?%@9Z?%-R*);5B"*#&%@1M.0%9Z?%-2*);.&/213.+A9:/?+^c5/?%#F.&/R13-.&/2%]^1$9{.@B/2% b?#0"5?]^%[_g+7.0/.&/?+A913b2b?#&">15*/+^9`.0/21M._`%:_g+,]^]p/21a8$%n.0"\.015 5%:1c5#0%1M.46%@13]" W .0#&"$Z??]^%:],+^-?+^-?c\Z?b.&/?%Fb2/2159&%9g9&".&/213. 46+A9&*"5->.&+^-=Z?+,.&+^%9g13#0%- .g+^-$.0#&"646ZR*)%4¡¢159g9&%%-+^-~£+,c$Z?#0%¤$B ¥ Concatenating Sinusoids of Different Frequency 1
Amplitude
0.5
0
−0.5
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (s)
£+,c$Z?#0%¤$¦ 9&+,c$-213][d1546%\=;*"5-2*1M.&%-21M.0+,-2c9&+,-=Z29&"5+A4?9" W 46+7§%#0%->. W #0%Y>Z?%-R*)+^%9_g+,]^] #0%9&Z?]7.:+^-z4?+^90*)"$-$.0+,-=Z?+,.&+^%9'+ W *15#&%+A9g-?"3.g.13 $%-.&"\[d1M.0*/.&/2%:+,-?+,.&+A13]b?/2159&%5B R %.Z21M.0+,"$- W "$#1\9.015-?%"$Z29`b?/2159&%" W 1*15#+,%#`9&+^c5-215] 13-R4+,.09 W #0%Y>Z?%-2*);_g+,]^]p%:["646Z2]^13.&%41$**)"$#04?+,-?c$],;$B
!
#"
%$'&
#"
(
)
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
£²+^c5Z2#&%:Ä?¦ 9&+,[\b?]^%F£!8>+^?#1M.0"+^-29.g+,-Í4B
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
z% *15-.&/?%#&% W "5#0%Z29&%£! 9&;=->.&/?%@9.0#&"$],]^+,-2c.&/?%µ¹ E MH¬S " W .&/?%8=+^?#1M.&"¯¡¬.&/?% 46%8>+A1M.0+,"$- W #&"$[ .&/?% *13#0#0+,%# W #0%Y>Z?%-2*);?¥15-24+7.9:L&¸MHKUB -+^-29.n*)#0%13.&%4+^-ÍD4¡O9&/?"M_g-+,-£+^c5Z?#0%:Ä$¥13]^]^"M_9 .&/2%Z29&%#.0"[\"646+ W ;µ.&/2%9&%b213#13[\%.&%#9B/?%_g+A4=.0/¯" W .&/2%8=+,?#1M.0"X+^946%).0%#0[\+,-?%@4 =;É.0/?%13[\b?]^+,.&Z246%" W .&/2% [\"646Z?]A1M.&+^-?cd9&+,c$-213]N?13-24.0/?%:#013.&%'" W 8=+,?#1M.0"\+^9g46%.&%#&[\+^-?%4=;d.&/2% W #&%@Y$Z2%-2*;" W .&/?%:[\"646Z?]A1M.0+,-2c\9&+^c5-215]¢B ¼K-"5#46%# W "5#n.&/?% %)§%*I.n.&"%Fb%#*)%+,8$%4159n8=+,?#1M.0"22.&/?%F8=+^?#1M.&"#013.&%:[ZR9{.R%F%]^"M_ .&/?%13Z24?+,?]^% W #0%Y>Z?%-2*); #13-?c$% 13-R4.0/?%\_g+^4=.0/[d1$46%dY$Z2+7.0%\9&[d13]^]¢B /?%-.&/2%8=+^?#1M.&"#1M.&%+A9+^-.&/?%d13ZR46+," W #0%Y>Z?%-2*);#13-?c$%513-R4_g/?%.&/2%:_g+^4=.0/+^9g[d1$46%']^15#&c$%#@$.0/?%'.&%@*/?-?+AY>Z?%:*13-%:Z29&%4.&"*)#0%13.&%'1\?#0"$154#015-?c5%" W 46+A9{.0+,-R*I.&+^85%'.0+,[ ?#0%9B
ge f8$%->.&%@4=; $"5/?- Ì /?"M_g-2+,-?c13.Ð>.13- W "$#04Ñ-2+,8$%#9.&%# W "5# Ì "$[ b?Z6.0%#©g%9&%15#0*/+,-!Z299&*+,]^]^13.&"$# W "5#%@15*/ W #0%Y>Z?%-R*);*"5[\b"5-?%-$.+,-.0/?%Fb?#0"646Z2*)%@499&*+,]^]^13.&"$#09+^-1d[\"5#0%:%)(*)+^%->.n_1a;d.0" +^-$.0#&"646ZR*)%F[\"5#0% W #0%Y>Z?%-2*);*)"$[\bR"$-?%->.09n+,-.&/?%\9Z?%-2*);\*)"$[b"5-2%->.09$+^- W #0%Y>Z?%-2*);[\"=4?Z?]^13.&+^"5-p61F9&+^c5-215] [\"646Z?]A1M.&%@9`.&/?% W #0%Y>Z?%-2*);" W 1*13#0#&+^%#">9&*+,]^]^13.&"$#B /?%'c5%-?%#13]%@Y$ZR1M.&+^"5- W "5#15-£! 9&"5Z?-R49&;=-$.0/?%9&+^Ï%#g+A9 ¡ >¥ ª ¡O«Z?%-R*);5Ê +A9'.&/?%d[\"646Z?]A1M.0+,-2c W #0%Y>Z?%-R*);5i15-24µ´ 13-R4É´ \13#0%13#0?+,.13#0;b?/R159&%*)"$-29.9B/?% W Z?-2*I.0+,"$-ɼ)¡¬.¥I`*15],]^%4.0/?%zº\J 6¶,¸3H¢EOJ3P ETP $U UP MU)¶,J ?U46%).0%#0[+^-?%@9n9&+,c$-?+ *13->.0],;d.0/?%F/215#&[\"$-?+^*:*"5->.&%->." W .&/?%9&"5Z2-24B£?#0"5[~¡ $¥`_`%:[d1a;46%.&%#0[\+,-2%:.&/213..&/2% +^-29{.13->.015-?%"$Z29 W #&%@Y>Z?%-2*;Ê À ¡¬«Z213->.0+7.{; ¡¬«.i" WW #0%Y>Z?%-R*);\46%8=+A1M.&+^"5-\b?#0"646Z2*%4d=; .&/?%[\"646Z?]A1M.0+,-2c"$90*)+^]^]^13.&"5#@B /2%- Á ?5.&/?%+,-R46% +A9g13]A9@?
A=
3>@?
A#
.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
)
1
0.5
0.5 J1(I)
J0(I)
1
0 −0.5
−0.5 0
5
10
15
20
−1
25
1
1
0.5
0.5 J3(I)
J2(I)
−1
0 −0.5 −1
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
0
0
5
10
15
20
−1
25
1
1
0.5
0.5
0 −0.5 −1
0
−0.5
J5(I)
J4(I)
0
0 −0.5
0
5
10
15
20
25
−1
£+,c$Z?#0%ËR¦i
`%909.&+^].0/?%:8M13]^Z?%'" W ¼D+A9g]^15#&c$%5B¼K-c5%-2%#13]N$.0/?%:/?+^c5/?%@9{.&N"$#04?%#0%49&+A46%215-24.&/213./2159g9&+,c$-?+ R*15-$.15[b2],+,.&Z24?%+A9gc$+,8$%=;d.&/2%F13b?b?#0" 6+,[d13.&%% 6b?#0%909¥'¡¢9@?
0
1
0 −0.5 −1
−0.5
1
−1
?
0
J−3(I)
J3(I)
−1
0 −0.5
0
5
10
15
20
−1
25
£+,c$Z?#&% ?¦i
%@9&9&%] W Z2-2*I.0+,"$-29g" W .&/?% R#09.9 " W .&/?%d9013[\% W #&%@Y>Z?%-2*;13#0% b2#&%@9.g+^9n1M. d 2B
" &
#
$
=
#
?Ê >_g/2%#0% \+A915-d+,->.&%c5%#Dc5#0%13.&%# .&/215-Ï%#&"RB/2% =
=
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
Magnitude (linear)
Spectrum of a simple FM instrument: fc = 220, fm = 110, I = 2 2500 2000 1500 1000 500 0 100
150
200
250
300
350
400
450
500
550
600
Frequency (Hz)
£+^c5Z?#0% ?¦DÐ=b%*).Z?[" W 1\9&+,[\b?]^%F£!~+^-29.2_g/?%#0%FÊ x¿5¿ 2?Ê ¤5¤ 2613-24 x¿6B
#
$#
Magnitude (linear)
Spectrum of a simple FM instrument: fc = 900, fm = 600, I = 2 2500 2000 1500 1000 500 0 0
500
1000
1500
2000
2500
3000
3500
Frequency (Hz)
£+^c5Z?#0% =¦DÐ=b%*).Z?[" W 1\9&+,[\b?]^%F£!~+^-29.2_g/?%#0%FÊ
2?Ê
##
2613-24 x¿6B
##
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
¼K-46%).0%#0[+^-?+^-?c\.&/2% W Z?-24?15[%->.013] W #0%Y>Z?%-2*);" W ;5"5Z2#£! 9&"5Z?-24p?+7.+A9Z29&% W Z?]p.&"46% 2-?%'.&/2% W "5]^],"M_g+^-?c#013.&+^"2¦ Ê ¡.&%c5%#09_g+,.&/-?"d*)"$[\["$- W 1$*I.0"5#9B /?% W Z?-R4?13[\%->.13] W #0%Y>Z?%-R*);+A9`.&/2%-c5+^85%-=; ʱ Ê Ê Ã ¡¥ È ¼K-£²+^c5Z2#&% W "5#% ?13[\b?]^%52Ê D¿5¿ 15-24Ê x ¤5¤ _g/?+A*/;>+^%]A4?9`.0/?%:#013.&+^"\" W ¿$¿ ¿ Ê Ê ¤$¤ ¤ È /?% W Z?-R4?13[\%->.13] W #0%Y>Z?%-R*);Êa±:+A9.0/?%#0% W "$#&%c5+^85%'=; ÊM± ¿5¿¿ ¤5¤¤ ¤$¤ p+^ 5%_g+^9&%'+,-£+^c5Z?#0% ==.&/2%:#013.&+^" " W Ê .&"Ê +A9Ä?¦ ¿13-24.&/2% W Z?-24?15[%->.013] W #&%@Y$Z2%-2*;+^9c$+,8$%-=; ʱ Ä ¿ Ä ¼W È _g/?%#& % +A9\13-É+^->.&%c5%#\c$#&%@1M.&%#F.0/213- ¤5 .0/?%- %8$%#0 ; .&/ /215#&[\"$-?+^*" W Ê ± +A9[\+^909.09¦%]^],N]^+, $%.&"5-2%9>_`"="64=¨4?#&Z?[6?#15909Z?%-2*);13. _g/?+A*/1 -2"3.&%'_g+^],]²9Z?%-2*;5B/?%#0% W "$#&%:+ W _`%b?]A1a;1-?"3.0% 13. {¿5¿ ?R.&/?% -?"3.0% 13-"=*).01a8$%g/?+,c$/?%#`_`"5Z?]A4\R% ¨Ë5Ë ?B +7.0/?+,-13-"6*).01a85%$3.0/?%#0%n13#0%F¤a¿'-?"3.0%9>.&/?%#&% W "5#0%_%*13-[\15 5%g.0/?% W "$],]^"M_g+^-?c #0%]A1M.&+^"5-R9_`%:[d1a; 2-24.0/?%:#&%]^13.&+^"5-29&/?+^b" W "$-?%'-?"3.0%F13"M85%:¿5¿ =; ¿5¿ ¿ Èn¿3Ä$Ä Rà p+^ 5%_g+^9&%5>.0/?% W #&%@Y>Z?%-2*;" W "5-2%-2"3.&%:%]^"M_ ¿$¿ Ï:+A9c5+^85%-=; ¿5¿ ¿ È x¿ 5Î 2à &"
"
#
#
@# #
#
@#
#
#
#
#
#
#
#
egf.091$9D.0/?%'bR%@13 69`+,-.&/2% 9&bR%@*I.13]%-=85%]^"5b%5B 913-\% ?15[b2],%$$b?#0"5-2"5Z?-2*%13]^"5Z24 .0/?%85"M_`%]A9 1$? %2 +^2 ">? Z2'_g/?+^],%n 5%%b?+^-?c'.&/?%9&15[\%gb?+7.*/ W "5#%15*/B Ð=+^-2*)%.&/?% b?+,.0*/+A9g.0/?%\9&15[%$R_`%F =-?"M_ .&/2% +^->.&%c$%##&%]^13.&+^"5-29&/?+^b" W .&/2% 9&bR%@*I.13]*)"$[b"5-2%->.09+^9.&/?%\9013[\%5Bn/2% W "5#0[d13->.09+A9_g/213.n13]^],"M_9`Z29`.0"d/?%15#14?+7§%#0%-2*%'R%.{_%%-.&/?%:8$"M_%]9. W #0%Y>Z?%-R*);+^913-+,->.&%c5%#n[Z2]7.0+,b?]^%" W .&/2% W Z?-24213[\%->.015]¢R.&/2% 9&bR%@*I.1\" W "3.0/*15#+,%#09_g+^]^]*"5[2+,-?%+,-9&Z2*/1d_1a;.0/?%F.0/?%*)"$[\bR"$-?%->.09n_g+,]^]"M85%#0]A13bp?15-24 1bR%@13 _g+,]^]R%F*#&%@1M.0%41M.g.&/2% W "5#0[\15->. W #0%Y>Z?%-2*);z¡O9&%%:£+,c$Z?#&%\¤ $¥IB @
"
)
#
#
¼K-É£+^c5Z?#0%¤ ?R"5.&/ *13#0#0+,%#09 13#0%[\"=4?Z?]^13.&%@4=;z.&/?%9&15[\%"$90*)+^],]A1M.0"5# _g+7.0/µ1 W #0%Y>Z?%-2*);Ê B/?%+^-246% X" W \[ "646Z?]A1M.&+^"5- W "5#`.&/2% 2#9.015] W #&%@Y>Z?%-2*;Ê ± .&/213. +A9g*)]^"$9&%9`.&".0/?%F46%9&+^#&%@4 W "$#&[d15-$. W #0%Y>Z?%-R*);È Ê¡ W #0"5[ iJMº 6HKUL =GE @¥IBi/R1M.+A9 Ê g ÊM± +^->.@¡¢Ê Êa±³ 2à $¥.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
Magnitude (linear) Magnitude (linear) Magnitude (linear)
FM spectrum with first carrier: fc1=400, fm=400, and f0=400 6000 4000 2000 0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
FM spectrum with second carrier: fc2=2000, fm=400, and f0=400 6000 4000 2000 0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
4500
5000
FM spectrum using first and second carriers 6000 4000 2000 0
0
500
1000
1500
2000
2500
3000
3500
4000
Frequency (Hz)
£+,c$Z?#0%¤ ?¦ /?%:9&bR%@*I.Z?[ " W +,-24?+,8=+A46+,ZR13]15-24*"5[ ?+,-2%4£! 9&+,c$-213]A9B #
ge fZ213].&" 2Bg£?"5#% ?13[\b?]^%5R_g/?%-Ê ¤ nÏ5Ê ¤:¤ Ï513-R4 Ê ¿ Ä Ï5R.&/2% 9&bR%@*I.13]²*)"5[\b"5-?%->. b?#0%9&%->.n+^-.&/2% 9&"5Z?-241M.Ë nÏ+A9.0/?% *"5[2+,-213.&+^"5-" W 9&+A46%215-24?9c5+^85%->;.&/2%b215+,#9¦ Ä2¾ ?¾ ¤ Ä ¾ ¿ R13-249&"d"5-X¡O9&%%F£+^c5Z?#0%\¤5¤@¥)Bi¼K- W 15*).6.0/?%#0%F13#0%:13-+^- 2-?+,.&%F-=Z?[ R%#" W ¾ 46;>154?9`.0/21M.b?#0"646Z2*% 19&+^46%213-R413. Ë Ï$B/?%1$*I.0Z213] -=Z?[ R%#:.&/213.*)"$-$.0#&+^?Z6.0%\.&".&/2%"M85%#015],] 13[\b?]^+,.&Z246%d+A9F46%.&%#&[\+^-?%4z=;.0% [\"646Z?]A1M.&+^"5-+^-246%*%9B
)
&
"
_
#"
)
#
$##
$# #
=
=
# #
@##
8=
#
8=
#
=
8=
##
!"646Z?]A1M.0+,"$-+^-246%*%9'13#0% 4?% 2-?% W "5#%1$*/*)"$[\bR"$-?%->.¦ F+^9.&/2% +^-246% .&/213.'*/215#013.&%#&+^Ï%@9g.&/2%9&bR%@*I.0#&Z?[ b2#&"5 46Z2*%4 =;'.&/?% 2#9{. [\"646Z?]A1M.0+,-?c"$90*)+^]^]^13.&"5#@M13-24 È +A9.&/213." W .&/2%9&%*"5-2424B/2%`15[b2],+,.&Z24?%`" W .0/?% 2 9&+^4?%215-24 ¡O® À ¥`+^9c$+,8$%-=;d.&/?%:b2#&"646Z2*)." W .&/?%F
`%909&%] W Z?-2*).&+^"5-29 ® À À ¡ ¥ ¡ È ¥IÃ ¡.&/?%-?%).g13[\b?]^+,.&Z246%" W 1*)"5[\b"5-?%->.`1M.13-=; W #&%@Y>Z?%-2*;\+^9D.&/?%*)"$[?+^-21M.0+,"$-" W [d15->; 9&+^46%213-R4?96_g/2%#0%-2%c$13.&+^85% W #0%Y>Z?%-R*)+^%9 W "$]^4?"M85%#:.0/?% Ï:?+^-X¡ iJMº 6HKUL =GIE @¥IB )
=
?
?
>
)
>
6=
> ?
=
=
>
#
]2
54
7
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B
Magnitude (linear)
FM spectrum with first modulator: fc=100 and fm1=100 6000 4000 2000 0
0
200
400
600
800
1000
1200
1400
Magnitude (linear)
FM spectrum with second modulator: fc=100 and fm2=300 8000 6000 4000 2000 0
0
200
400
600
800
1000
1200
1400
Magnitude (linear)
FM spectrum with two modulators: fc=100, fm1=100, and fm2=300 4000 2000 0
0
200
400
600
800
1000
1200
1400
£+,c$Z?#0%¤$¤5¦ /?%F£! 9Z?%-2*);*)"5[\b"5-?%->.09B
Ì ! ÍDxÄ2¤Î2¦£2Z?-24?15[%->.013]A9g" W Ì "5[\b?Z?.&%#0+^Ï%4Ð="$Z?-24?2 Ì #&%@1M.0%415-24[d13+^-$.13+^-?%4=;13[d13#1Ð6[;>.&/pÐ6*/?"="5] " W Ì 5" [\b?Z?.&+^-?cÐ6*)+^%-2*%5RÐ6+,[\"5-£?#159&%#Ñn-?+^85%#9&+7.{;$B