! y n[ ]= sin(2"#350# n)T=?? samples [integer]! y n[ ]= y[n + T]50/3 ! integer! y n[ ]= sin(2"#350# n)T=?? samples! y n[ ]= y[n + T]! sin(0) = sin(2"k)k=1,2…! 2"350# n = 2"k! nk=503samplescycleRatio ofintegersrational numberT=n=50 samples, k=3 cyclesperiodic! y n[ ]= sin(2"#225# n)T=?? samples! y n[ ]= y[n + T]! sin(0) = sin(2"k)k=1,2…! 2"225# n = 2"k! nk=25 22irrational numberEquiv. discrete sinusoid not periodic0 0.5 1 1.5 2 2.5 3 3 .5 4-1-0.8-0.6-0.4-0.200.20.40.60.81time (sec)y=sin(2*pi*sqrt(2)/25*n)TextEndirrational frequency! y t( )= sin(2"# 2 # t)! T =12seccontinuous functionperiodicTs=1/25 secPeriod of Sum of Sinusoids! y t( )= y t + T( )0 1 2 3 4 5 6-1-0.500.510 1 2 3 4 5 6-2-1012time (sec)Tsum=3 secondsT1=0.2 seconds, T2=.75 secondsLeast common multipleTsum=3 secondsT1=1/5 seconds1/5*k=3/4*lk/l=15/44/20s, 8/20s, 12/20s,16/20s, 20/20s, 24/20s,28/20s, 32/20s, 36/20s,40/20s, 44/20s, 48/20s,52/20s, 56/20s, 60/20s15/20s. 30/20s,45/20s, 60/20s 15 cycles4 cycles1/5s, 2/5s, 3/5s …T2=3/4 seconds3/4s, 6/4s, …Tsum=15*T1=15/5=3 secondsTsum=4*T2=3/4*4=3 secondsseconds to complete cyclesseconds to complete cyclesrational numberAddition cartesianPowerspolarRootspolarpolarcartesian polar cartesians=a+jb s=rej!! s = a2+ b2ej"a tanba( )! s = r cos"+ jr sin"! a1+ jb1( )+ a2+ jb2( )= a1+ a2( )+ j b1+ b2( )Subtraction cartesianMultiplicationpolarDivisionpolar! r1ej"1# r2ej"2= r1r2ej"1+"2( )! a1+ jb1( )" a2+ jb2( )= a1" a2( )+ j b1" b2( )! r1ej"1r2ej"2=r1r2ej"1#"2( )! rej"( )n= rnejn"! zn= s = rej"! z = s1/ n= r1/ nej"/ n +2#k / n( ) ! k = 0,1,2Kn "1Complex ArithmeticComplex ConversionsAddition cartesianPowerspolarRootspolarpolarcartesian polar cartesianSubtraction cartesianMultiplicationpolarDivisionpolar! z = 641/ 3ej 0 / 3 + 2"k / 3( )= 4 ej 2"k / 3( )Complex ArithmeticComplex Conversions! 2ej"#3= 2cos#3+ j2sin#3= 1 + j 3! 3 + j4 = 32+ 42ej"a tan43( )= 5ej"0.927! 1+ j2( )" 3 + j4( )= "2 " j2( )! 1+ j2( )+ 3 + j4( )= 4 + j6( )! 5ej"#3" 6ej"#4= 5 " 6ej"#3+#4( )= 30ej"7#12! 10ej"#2÷ 5ej"#4=105( )ej"#2$#4( )= 2ej"#4! 3ej"#4( )3= 33" ej"3#4( )= 27ej"3#4( )! z3= 64 = 64ej 0! 44ej 2"/ 3( )4ej 4"/ 3( )polarRoots! zn= s = rej"! z = s1/ n= r1/ nej"/ n +2#k / n( ) ! k = 0,1,2Kn "1! 4 = 41 2! 4! 2! 2 " 2 = 4! "2! "2 #"2 = 4! 4ej0( )1/ 2= 41/ 2ej 0 /2+2"k /2( )! = 2ej 2"/2( )! = 2ej 2"0 / 2( )! = 2ej"! = 2 "#1! = 2ej0! = 2 "1! = "2! = 2! 4 = ±2! k = 0! k =1we expect n=2 solutionsr=2ReImpolarRoots! zn= s = rej"! z = s1/ n= r1/ nej"/ n +2#k / n( ) ! k = 0,1,2Kn "1! 2! 2 " 2 "2 " 2 = 16! "2! "2 #"2 #"2 #"2 = 16! 16ej0( )1/ 4= 161/ 4ej 0 /4+2"k / 4( )! = 2ej 2"1/ 4( )! = 2ej 2"0 / 4( )! = 2ej"/2! = 2 "1 j! = 2ej0! = 2 "1! = j2! = 2! 16 = ±2,± j2! k = 0! k =1we expect n=4 solutions! 164= 161 4! 44= 41 4! ?! ?! = 2ej 2"3/ 4( )! = 2ej 2"2 / 4( )! = 2ej 3"/2! = 2 "#1 j! = " j2! = 2! k = 2! k = 3! = 2ej"! = 2 "#1r=2ReIm! " j2 #" j2 #" j2 #" j2 = "2 #"2 = 4! Acos 2"kf0t +#k( )! Re Aej 2"ft +#{ }! = Re Aej"ej 2#ft{ }! = Re Xej 2"ft{ }! Aej"#ej 2$ft+e% j 2$ft2( )! = X "ej 2#ft+e$ j 2#ft2( )Representations of Sinusoids! y n[ ]= cos(2"#18# n)! y(t) = cos(2"t)Discrete SinusoidContinuous Sinusoidn012345678!(rad)0"/4" /23 " /4"5 " /43 " /27 " /42 "0 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0 .9 1-1-0.8-0.6-0.4-0.200.20.40.60.81! "= 2#t! "=#4n! y(t) = cos(")! y n[ ]= cos(")t(sec)01/81/43/81/25/83/47/81cos(!)10.7070-0.707-1-0.70700.7071! A cos 2"ft +#( )! cos 2"t( )! Ts=18sec! t = nTsSample rate:! y n[ ]= cos(2"#18# n)! y(t) = cos(2"t)Discrete SinusoidContinuous Sinusoidn012345678!(rad)0"/4" /23 " /4"5 " /43 " /27 " /42 "0 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0 .9 1-1-0.8-0.6-0.4-0.200.20.40.60.81! "= 2#t! "=#4n! y(t) = cos(")! y n[ ]= cos(")t(sec)01/81/43/81/25/83/47/81cos(!)10.7070-0.707-1-0.70700.7071exp(j!)1+0j0.707+j0.7070 + 1j-0.707+j0.707-1+0j-0.707-j0.7070+j10.707-j0.7071+j1! = Re Aej2"ft+#{ }! = Re Aej"ej2#ft{ }! = Re Xej2"ft{ }! phasor! X = Aej"! complexamplitude! A cos 2"ft +#( )! cos 2"t( )! = Re ej2"t{ }! X = 1ej0= 1! Ts=18sec! t = nTsSample rate:Rotating Phasor0 0 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.8-0.6-0.4-0.200.20.40.60.81! y n[ ]= cos(2"#18# n +15"180)! y(t) = cos(2"t +15"180)! Ts=18sec! t = nTsDiscrete SinusoidContinuous SinusoidSample rate:n012345678!(rad)0.2621.0471.8332.6183.4034.1894.9745.7606.545! "= 2#t +15#180! "=#4n +15#180! y(t) = cos(")! y n[ ]= cos(")t(sec)01/81/43/81/25/83/47/81cos(!) 0.966 0.500-0.259-0.866-0.96-0.500 0.259 0.866 0.966exp(j!) 0.966-0.259j 0.500-j0.866-0.259+j0.966-0.866-j0.500-0.966+j0.259-0.500+j0.866 0.259+j0.966 0.866+j0.500 0.966-j0.259! = Re Aej2"ft+#{ }! = Re Aej"ej2#ft{ }! = Re Xej2"ft{ }! phasor! X = Aej"! complexamplitude! A cos 2"ft +#( )! cos 2"t +15"180( )! = Re ej 2"t+15"180( )# $ % & ' ( ! X = 1ej15"180= ej15"180! = Re ej15"180ej2"t# $ % & ' ( ! = Re Xej2"t{ }Rotating Phasorw/ initial phase! y n[ ]= cos(2"#18# n)! y(t) = cos(2"t)Discrete SinusoidContinuous Sinusoidn012345678!(rad)0"/4" /23 " /4"5 " /43 " /27 " /42 "0 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0 .9 1-1-0.8-0.6-0.4-0.200.20.40.60.81! "= 2#t! "=#4n! y(t) = cos(")! y n[ ]= cos(")t(sec)01/81/43/81/25/83/47/81cos(!)10.7070-0.707-1-0.70700.7071exp(j!)1+0j0.707+j0.7070 + 1j-0.707+j0.707-1+0j-0.707-j0.7070+j10.707-j0.7071+j1! = Aej 2"ft+#( )+e$ j 2"ft+#( )2% & ' ( ) * ! = X "ej2#ft+e$ j 2#ft2% & ' ( ) * ! X = Aej"! Ts=18sec! t = nTsSample rate:! A cos 2"ft +#( )! cos 2"t( )! complexamplitude! = 1"ej 2#t+0( )+e$ j 2#t+0( )2% & ' ( ) * ! = X "ej2#t+e$ j 2#t2% & ' ( ) * ! X = 1ej0= 1exp(-j!)1-0j0.707-j0.7070 - 1j-0.707-j0.707-1-0j-0.707+j0.7070-j10.707+j0.7071-j1complex conjugate pairsSum multiple cosines same frequency! Acos 2"kf0t +#k( )! Re Aej 2"ft +#{ }! = Re Aej"ej 2#ft{ }! = Re Xej 2"ft{ }! Aej"#ej 2$ft+e% j 2$ft2( )! = X "ej 2#ft+e$ j 2#ft2( )! Akcos 2"ft +#k( )k=1n$! = Re Ake2"ft +#k{ }k=1n$= Re Ake#ke2"ft{ }k=1n$! = Re Ake"k{ }k=1n#$ % & ' ( ) e2*ftEx.Representations of Sinusoids! 3cos 2"40t +"2( )#1cos 2"40t #"6( )+ 2 cos 2"40t +"3( )! 3ej"2ej 2"40t#1e# j"6e2"40t+ 2e"3e2"40t! 3ej"2#1e# j"6+ 2e"3( )e2"40t! 5.234ej1.545e2"40t! 5.234 cos 2"40t +1.545( )multiply cosines of different frequency! A1cos"1t( )# A2cos"2t +$( )! A1ej"1t+ e# j"1t2$ % & ' ( ) A2ej"2t +*( )+ e# j"2t +*( )2$ % & ' ( ) ! A1A24ej"1tej"2t +#( )+ ej"1te$ j"2t +#( )+ e$ j"1tej"2t +#( )+ e$ j"1te$ j"2t +#( )( )! A1A24ej"1t +"2t +#( )+ e$ j"2t$"1t +#( )+ ej"2t$"1t +#( )+ e$ j"1t +"2t +#( )( )! A1A22cos"1+"2( )t +#( )+ cos"2$"1( )t +#( )( )Composite signals (waveform synthesis)! x(t ) = A0+ Akk=1"#cos 2$fkt +%k( )= X0+ Re Xkk=1"#ej