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.     � [ What do you do with negative amplitudes? Period of Sum of SinusoidsAmplitudes always positive C cos 2ft = C cos 2ft ±( ) ( ) 2 complex amplitude spectrum Absorb the minus sign into the angle 1.8 1.6 3cos 2� 5t + 3 = 3cos 2� 5t +( ) ( 3 ) xt()= cos 25t( )1.4 1.2 = 3cos 2� 5t 2( 3 ) TextEnd 1.5e j 2 3 1.5e j 2 3 complex amplitude - X1 0.8 0.6 0.4 0.2 j 2ft +� j 2ft +�e + eyt = cos 24 t-6 -4 -2 0 2 4 6 () ()( 3)cos 2ft +� = 0 frequency - f ( )2 j�j 2ft j� j 2fte e + e eamplitude spectrum = 2 TextEnd 1.5 1.52 1.5 = Xkme j 2ft + Xkpe j 2ft Xkm =12 ej� amplitude - A1 0.5 = �Xke j 2ft 0 -6 -4 -2 0 2 4 6 zt()= xt()+ yt() � 12 ej� k = f 3 TextEnd 2 3 phase spectrum zt()= zt(+ T),T = ? �12 e � j� k = � f 2 2 3 X = 1  0 otherwise phase - phi0 -1 -2 � always a odd pair -3 -6 -4 -2 0 2 4 6 frequency - f Least common multiple ()(()) Fourier Series xt()= cos 2(5t) yt = cos 24 t311 t 02. . Tsin 2�t) 0 � t <1seconds to complete cycle x(t) =( sin seconds to complete cycle T=3/4 seconds �� �� �� 0 0.5 1 1 1y= 0 +1� cos 2t + 0 � cos 22t + 0 � cos 23t +KTx=1/5 seconds 0 t 1� 2 � � 2 � � 2 � 3/4s, 6/4s, … X0 = 01/5s, 2/5s, 3/5s … X�1e� j�/2 k =1 k =�4/20s, 8/20s, 12/20s, 15/20s. 30/20s, � 0 k � 1 16/20s, 20/20s, 24/20s, 28/20s, 32/20s, 36/20s, 45/20s, 60/20s X0 = 1 T� 0 x(t)dt 040/20s, 44/20s, 48/20s, 4 cycles T0 1 52/20s, 56/20s, 60/20s 1/5*k=3/4*l =� sin(2t)dt = 0 0 15 cycles k/l=15/4 rational number 2 T0 � j2kt Xk =� x(t)e Todt zt()= xt()+ yt() T00 1 � j2Tz=15*Tx=15/5=3 seconds Tz=4*Ty=3/4*4=3 seconds = 2 sin(2�t)ektdt 0 zt()= zt(+ T) 1 z = 2 sin(2t) cos(�2kt) + j sin(�2kt)]dt 0Tz=3 seconds1 Fourier Series Fourier Series (frequency space) T1 .. . . tj2  k . T2cos(2kt) sin(2t)cos(2kt) � sin(2t)cos(2kt)dt 1 . .Tz te dt 0 z t dt Xk T 0sin(2t) sin(2kt) sin(2t)sin(2kt)0 � 1sin(2t)sin(2kt)dt Xk X0 T 30 3 t . . . .j2  k1 4 . 2.. .tcos 2..5 t. cos d t 2 .X0 cos 2..5 t. cos 2..4.t .e 3dtXk3 3 3 30 0 = 0 0 =1e j/2 X0 2 exp 2i  kk3 2k3 241 exp 2i  kkk = 1 . . . . . .. . . . . .241 k .iXk = 0.5 .k4 241..k2 3600. k�4 , 15 k4, k15 21k3 2k3 2411k = 0 . ... . .241 k 0 Xk 1i. . . . . k 15 k 4 k 4 k 15 415 = 0 X,0 0Xkk = 2 = 0 So, use L’Hopitals Rule = 0 Xk 1 ... . . . . . . . . . . . .... . . . .41i  exp 21i  kk3 6 exp 21i  kk2 6k2 482i..exp 21i  kk 241 exp 21i  k 241 .1i. 4 .k 4 k . . . .. . .4 k 15 .k 15 k 4 k 15 .k 15 k 4 k 15 .k 15 k 4 k 4 k = 3 = 0 1 = 0 X15 X4 1 Fourier Series (frequency space) Aperiodic Sum of Sinusoids w/ an Irrational Frequency Spectrum of z=cos(2pi*5t)+cos(2pi*(4/3)t) 0 1X0 aperiodic sum of sinusoids 1 period:3/4 secondsperiod:5sqrt(2) seconds0.90 k�4 , 15 Xk 0.8 X 14 0.7 X15 0.6 0.5 xt = cos 252t()( )0 -0.5 -1 0 1TextEnd2 3 4 5 6 1 0.5 yt = cos 24 t() ()( 3) 0 -0.5 0.5TextEnd 0.4 X0.3 0.2 0.1 0 TextEnd-1-20 -15 -10 -5 0 5 10 15 20 frequency 0 1 2 3 4 5 6 2 period:?seconds1 0 zt()= xt()+ yt() -1 TextEnd zt()= zt(+ T),T = ?-2 0 1 2 3 4 5 6 timex 52 Fourier Series for Irrational Frequency xt()= cos 252tyt = cos 243t() Least common multiple ()()()“What’s the frequency, Kenneth?” Aperiodic sum of sinusoids 1 seconds to complete cycle In 1986, CBS Anchorman Rather was confronted about 11 p.m. while walking onseconds to complete cycle 0.5 Park Avenue, when he was punched from behind and knocked to the ground then 1 0T = seconds 3 chased into a building and kicked him several times in the back while the assailant-0.5 T seconds52 =-1 TextEnd y2 3 4 5 6 4 demanded to know 'Kenneth, what is the frequency?' The assailant was convinced 0 1 1 0.5period:3/4 secondsperiod:5sqrt(2) seconds1 2 3 0 the media had him under surveillance and were beaming hostile messages into3 6 9s, s, sK -0.552 52 -1 TextEnd s, s, sK 0 1 2 3 4 5 6 2 4 4 4 1 0 -1 TextEnd his head, and he demanded that Rather tell him the frequency being used. period:? secondsIn the Fourier Series for an aperiodic signal -2 0 1 2 3 4 5 6 time 1 3 “what’s the period, Quinn?” 0.60.410218k = l 524 T T . ...t 1 1j 2  k 0.42 z t dt .. A k k 15 2 X0T. XkT z te Tdt 20= irrational number 0 0.2l 4 3 .5.252191 10 0 zt()= xt()+ yt() PS3-5 40 20 0 20 40 50 k 50 zt = zt + T()()zPick a T, plot the spectrum, then repeat with larger T’s. T = �seconds zt() aperiodicz Bases are building blocks to form more complex things QUAD8 Numerically evaluate integral, higher order method. Q = QUAD8('F',A,B) approximates the integral of F(X) from A to B 'F' is a string containing the name of the function. Synthesize x from a weighted sum of basis elements, � m The function must return a vector of output values given a vector of input values. x =�Ak�k orQ = QUAD8('F',A,B,TOL,TRACE,P1,P2,...) allows coefficients P1, P2, ... to be passed directly to function F: G = F(X,P1,P2,...). k=1 Decompose x into a weighted sum of basis elements, � To use default values for TOL or TRACE, you may pass in the empty matrix ([]). Basis Vector 1 x(t) =� cos2 ()2ft dt V1 V10 function y = myintegrand(t,f) V V2 y=cos(2*pi*f*t).^2; V2 V save as myintegrand.m x2 x1 V = axˆ1 + bxˆ2 x2 V = axˆ1 + bxˆ2»f=1; x1 »quad8('myintegrand',0,1,[],[],1) ans = Prefer orthonormal basis vectors 0.50000000000000 T V V•x2 VT . . . .t 2X0 T1.0 Xk T 0 z te1j 2  k Tdt V•x2 projectionsz t dt . . Pick a T, compute X0, loop over Xk’s, plot the spectrum, then repeat with larger T’s. …