MIT OpenCourseWare http://ocw.mit.edu MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Period of a Discrete Sinusoid yn = sin(2 1  n)[]50 T=50 samples yn = y[n + 50][]sin(0) = sin(2 ) yn = sin(2 50  n) 50  n = 2 k[]32 3 yn = y[n + T] T=?? samples n = 50 samples Ratio of []sin(0) = sin(2 k) k 3 cycle integers k=1,2… rational number periodic T=n=50 samples, k=3 cycles yn[]= sin(2 3  n)50 yn = T] T=?? samples [integer] 50/3 � integer []y[n +irrational frequency1 0.8 0.6 TextEnd yt()= sin(2  2  t)0.4 0.2 T =1 sec 2 0 continuous function -0.2 periodic-0.4 -0.6 -0.8 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) Ts=1/25 sec yn[]= sin(2 252  n)2 2  n = 2 k25 yn = y[n + T] T=?? samples []n 25 2 = sin(0) = sin(2 k) k=1,2… k 2 Equiv. discrete sinusoid not periodic irrational number y=sin(2*pi*sqrt(2)/25*n)Period of Sum of Sinusoids yt = yt + T() () Least common multiple seconds to complete seconds to complete cycles cycles T1=1/5 seconds T2=3/4 seconds 1/5s, 2/5s, 3/5s … 3/4s, 6/4s, … 4/20s, 8/20s, 12/20s, 16/20s, 20/20s, 24/20s, 28/20s, 32/20s, 36/20s, 15/20s. 30/20s, 45/20s, 60/20s 40/20s, 44/20s, 48/20s, 4 cycles 52/20s, 56/20s, 60/20s 1/5*k=3/4*l 15 cycles k/l=15/4 rational number Tsum=15*T1=15/5=3 seconds Tsum=4*T2=3/4*4=3 seconds Tsum=3 seconds 1 0.5 0 -0.5 -1 0 1 3 5 2 4 6 T1=0.2 seconds, T2=.75 seconds 2 1 0 -1 -2 0 1 2 3 4 5 6 time (sec) Tsum=3 seconds Complex Conversions cartesian polar polar cartesian ()s=a+jb s = a2 + b2 ej� a tan bas=rej� s = r cos�+ jr sin� Complex Arithmetic Addition cartesian jb1+ jb2= + jb1 +Subtraction cartesian ()()()a1 + a2 +()a1 +()()()a2b2jb1� jb2= + jb1 � b2() a1 + a2 + a1 � a2() Multiplication polar r1ej�1  r2ej�2 = r1r2ej �1 +�2r1ej�1 r1 ()Division polar j�2 = ej �1 ��2r2e r2 Powers polar re j�n =ne jn�()rRoots polar zn =1/ sn = re 1/ j� n()2 k = 1, 2 K n �1j � / n + k / nz = s = r e� � � � �  � � � � � � � � � � �Complex Conversions Representations of Sinusoids cartesian polar polar cartesian Acos 2 kf0t +Re Ae j 2 ft +� Ae j��(ej 2 ft +e � j 2 ft )(�k ) { } 2 ()3 = Re{Ae j�ej 2 ft }= X �(ej 2 ft +2 e � j 2 ft )32 42 j� a tan 43= 5ej� 0.927 2ej 3 = 2cos 3 + j2sin =1+ j3 + j4 = + e 3 = Re{Xe j 2 ft } Sum multiple cosines same frequency n n n cos 2�ft +�{ 2�ft +�k }�{ �ke2�ft }�Ak (�k )= Re Ake = Re Akek=1 k=1 k=1 � n  =�Re{Ake�k }e2�ft � k=1  Ex. 3cos 2 40t + �1cos 2 40t � + 2 cos 2 40t + 3( 2 ) ( 6 ) ( ) � j 2 ej2 40t � j 6 ej2 40t 3 ej2 40t � Re 3� e �1e + 2e � �� j � j � j2 40t � Re�3e 2 �1e 6 + 2e 3 e  Re 5.234{ ej1.545 e 2 40t } 5.234 cos 2 40t +)( 1.545 Composite signals (waveform synthesis) Complex Arithmetic Addition cartesian 1+ j2( )+ 3 + j4( )= 4 + j6( ) Subtraction cartesian 1+ j2( )� 3 + j4( )= �2 � j2( ) Multiplication polar 5e j 3  6e j 4 = 5  6e j 3 + ( 4 )= 30e j7 12 Division polar 10e j� 2 ÷ 5e j� 4 =10 5()e j� 2 � ( 4 )= 2e j� 4 Powers polar 3e j 4( ) 3 = 33  e j3 4()= 27e j3 4() Roots polar z = 641/ 3 e j 0/ 3+2 k /3( )= 4e j 2 k /3( ) z3 = 64 = 64e j 0 4 4e j 2 ( /3) 4e j 4 ( /3) multiply cosines of different frequency Ak cos 2�kf0t + = Re(�k )X0 +k=1 Xke j 2�kf0tx(t) = A0 + A1cos �1t � A2cos �2t +� k=1() ( ) A1� ej�1t + e� j�1 t  A2�ej(�2 t +�)+ e � j(�2t +�) decompose a periodic signal x(t) into a sum of a series of � 2 � 2  sinusoids - the Fourier series. A1A2 (ej�1 te j(�2 t +�)+ ej�1te � j(�2t +�)+ e� j�1te j(�2 t +�)+ e� j�1te � j(�2t +�))4 Note: The sum of periodic functions is periodic. A1A2 (ej(�1 t +�2t +�)+ e � j(�2 t��1t +�)+ ej(�2 t��1t +�)+ e � j(�1t +�2t +�))4 ex.A12 A2 (cos((�1 +�2 )t +�)+ cos((�2 ��1)t +�)) Xk = ��� �28 k2 k odd � 0 k even f0 = 25Hz� � � � � � � = � � � � �� � �� � � � � � � � � � �� � �� � � � � � � � � � Composite signals (waveform synthesis) Composite signals (waveform synthesis) Xke j 2�kf0tx(t) = A0 + cos 2�kf0t +�k )= X0 + Re�Ak ( Xke j 2�kf0t x(t) = A0 + cos 2�kf0t +�k )= X0 + Re�Ak ( k=1k=1 �k=1 � k=1 �8 k odd Xk = 8 2k2 e j k odd = 8 e j k oddXk 2k2 Xk 2k2 �0 k even 0 k even 0 k even k=1 k=3 x(t) = 0.8105cos 2 25t + x(t) = 0.8105cos 2 25t + )+ 0.0901cos 2 75t + ( ) ( ( ) 1 1 1 10.80.8 0.8 0.80.60.6 0.6 0.60.40.4 0.4 0.40.20.2 0.2 = 0.2 0 = 0 00 -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -0.8-1-1 0 0.02 0.04 0.06 0.08 0.1 0.120 0.02 0.04 0.06 0.08 0.1 0.12 -1 0 0.02 0.04 0.06 0.08 0.1 0.12-1 0 0.02 0.04 0.06 0.08 0.1 0.12 1 0.8Composite signals (waveform synthesis) 0.6 Xke j 2�kf0 t spectrum 0.4 x(t) = A0 + cos 2�kf0t +�k )= X0 + Re�Ak ( k=1 k=1 0.2 0 0.45 -0.2 0.4053e� j j -0.48 ej k odd 0.4053eXk = ��� �� 0.4 2k2 -0.6 -0.80 k even 0.35 -1 0 0.02 0.04 0.06 0.08 0.1 0.12k=5 0.3 x(t) = 0.8105cos 2 25t + )+ 0.0901cos 2 75t + )+ 0.0324 cos 2 125t + ( ( ( ) X 0.25 1 1 0.2 0.80.8 0.60.6 0.15 0.40.4 = 0.20.2 0.1 0.0450e� j 0.0450ej 0 -0.2 0 0.05-0.2 0.0162e � j 0.0162ej -0.4 -0.4 -0.6 0 300f-300 -200 -100 0 100 200 -0.8 -125 -1 0 0.02 0.04 0.06 0.08 0.1 0.12 x(t) = 0.8105cos 2 25t + ( -75 -25 25 75 125 )+ 0.0901cos 2 75t + ( )+ 0.0324 cos 2 125t + )+ ...(-1 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.8 -0.6� � � � � � � � � � � � � � � � …