MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology Fall 2007

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1 MIT OpenCourseWare MAS.60 / MAS.50 / MAS.5 Signals, Systems and Information for Media Technology Fall 007 For information about citing these materials or our Terms of Use, visit:

2 Linearity input Ax (t)+bx (t) Time invariance Linear Time Invariant Systems x(t) Linear y(t) Time Invariant System scaling & superposition output Ay (t)+by (t) x(t- ) y(t- ) Characteristic Functions e st s=a+jb complex exponentials s=a+jb e st Complex Exponential Signals s= e - t exponential decay s=±j e ±j t sinusoids s=- ±j e - ±j t exponential sinusoids e cos = j + e j characteristic functions of LTI systems x(t)=sin( t) Linear Time Invariant System y(t)=a sin( t+ ) output has same frequency as input but is scaled and phase shifted Periodic y( )=y( + n) sine odd sin(- )=sin( ) Sinusoids (rad) y =sin( ) y =cos( ) 0 0 / / / / / 0 0 cosine even cos(- )=-cos( ) Relations: = f T:period (sec/cycle) Continuous sinusoids T = /f= / = (t) y(t)=a sin( t+ ) y(t)=a sin( ft+ ) Parameters: A: amplitude : phase (radians) : radian frequency (radians/sec) or f: frequency (cycles/sec-hz) rad/sec= ( rad/cycle)*cycle/sec y(t)=y(t+t) sec/cycle= / (cycle/sec)= ( rad/cycle)/(rad/sec)

3 Continuous sinusoids = (t) Sampled Continuous Sinusoid y(t)=a sin( t+ ) y(t)=a sin( ft+ ) Continuous Sinusoid sampled continuous sinusoids Parameters: y(t) = sin( t) A: amplitude Sample rate: T = 50 sec s 0 : phase (radians) -0. TextEnd : radian frequency (radians/sec) t = nt s -0.4 or -0.6 f: frequency (cycles/sec-hz) Discrete Sinusoid -0.8 Phase shift yn [ ]= sin( n) In: x(t)=sin(t) x(0)=0 yn sin( ) n y[n] [ ]= 5 n 0 0 Out: y(t)=0.5sin(t- /3) y( /3)=0 y[n]=cos(*pi*n/50) n - samples Discrete sinusoids = [n] n=0,, Period of discrete sinusoids y[n]=a sin( n+ ) y[n]=a sin( fn+ ) A: amplitude : phase (radians) : radian frequency (radians/sample) f: frequency (cycles/sample) Relations: = f rad/sample= ( rad/cycle)*cycle/sample N:period (samples/repeating cycle [integer]) Smallest integer N such that y[n]=y[n+n] Find an integer k so N=k/f is also an integer N f f = k/n (f: rational number -> k/n is ratio of integers) ex: y[n]=cos( (3/6)n) 0.6 What is the period N? 0.4 y[n]=a cos( fn+ ) frequency: f=3/6 cycles/sample y[n]=cos(*pi*3/6*n) TextEnd period of discrete sinusoids -0.6 N:period -0.8 (samples/repeating cycle [integer]) Smallest integer N such that y[n]=y[n+n] Find an integer k so N=k/f is also an integer f=3/6 let k=3 N=(3)*6/3=6 n - samples f = k/n (rational number -> k/n is ratio of integers)

4 [ ] sin( 3 yn = 50 n) f = 3 cycles 50 sample yn [ ] = y[n +N ] N=?? samples N: integer N f 50/3 integer Period of discrete sinusoids: ex. discrete function yn [ ] = sin( 3 50 n) frequency: f = 50 3 Period of discrete sinusoids: ex. cycles sec period: N=?? samples yn [ ]= y[n + N ] f N = k 3 50 N = k N, k:integers N 50 samples ratio of integers periodic = k 3 cycle rational number N=50 samples, k=3 cycles Aperiodic discrete sinusoids continuous function yt ()= sin( t) 0.4 T = sec periodic 0. sample t = nt s T s = 5 sec y=sin(*pi*sqrt()/5*n) TextEnd irrational frequency arbitrary continuous signal y(t+ )=y(t) After what interval does the signal repeat itself? Periodicity T:period (sec/cycle) -0.8 discrete function - [ ] yn = sin( 5 n) period? [ ] y[n + yn = N ] N=?? samples (integer) time (sec) f = 5 f N = k N,k: integers N 5 not a ratio of integers = k irrational number sampled discrete sinusoid aperiodic

5 Ex: Period of sum of sinusoids T=0. seconds, T=0.75 seconds seconds to complete cycles T=/5 seconds /5s, /5s, 3/5s Least common multiple seconds to complete cycles T=3/4 seconds 3/4s, 6/4s, 4/0s, 8/0s, /0s, 5/0s. 30/0s, 6/0s, 0/0s, 4/0s, 45/0s, 60/0s 8/0s, 3/0s, 36/0s, 40/0s, 44/0s, 48/0s, 4 cycles 5/0s, 56/0s, 60/0s /5*k=3/4*l 5 cycles k/l=5/4 rational number - Tsum=5*T=5/5=3 seconds Tsum=4*T=3/4*4=3 seconds time (sec) Tsum=? seconds Tsum=3 seconds y( )=sin( ) = (t) instantaneous frequency =d /dt sinusoid constant frequency Instantaneous frequency y(t)=a sin( t+ ) = t+ d /dt= chirp linearly swept frequency =d /dt =(( - 0 )/T)t + 0 integrate =( - 0 )/T t + 0 t+c time varying argument t 0 0 y chirp (t)=a sin(( - 0 ) /(T) t + 0 t+ ) Representations of a sinusoid y(t)=a cos( t+ ) y(t) = Ae j trig function [ e j t + e j t] j y(t)=re{ae j e j( t) } X= Ae j complex amplitude (constant) y(t)=re{xe j( t) } complex conjugates e ( ) = e j( t+ ) realpartof complex exponential rotating phasor Euler s relations e j =cos( )+jsin( ) cos = ej + e j sin = ej e j j j = e e j t

6 Complex Exponentials Why use complex exponentials? Trigonometric manipulations -> algebraic operations on exponents Trigonometric identities cos(x)cos(y) = [cos(x y) [cos(x + y)]] ( ) cos + cos x (x) = (re Vector representation (graphical) Properties of exponentials re x e y =re x+y x ) n =r n e nx n / n x = x = x x Complex Exponentials Amplitude modulation (multiply two sinusoids of different frequencies) Acos( t) B cos( t+ )= C(cos( 3 t+ )+ cos( 4 t+ )) *sum formula A. cos. B. cos for sin & cos A. B. cos. cos. cos sin. sin trig id A. B. cos. cos A. B. cos. sin. sin *product formula cos cos sin sin for sin & cos AB...cos AB...sin trig id cos s cos d sin s sin d AB...cos AB...sin *sum formula. AB.. cos. cos s cos. cos d sin. sin s sin. sin d for sin & cos. AB.. cos s sin d trig id or A. cos. B. cos j. j. j. j. e e e e cos = complex conj A.. B. mult. exponentials. AB.. exp j. exp j. exp j. exp j. 4. AB... cos. cos cos = complex conj 4. AB.. cos cos Representations of a sinusoid y(t)=a cos( t+ ) trig function e j t + e j t j j t+ y(t) = Ae j [ ] complex conjugates e ( ) = e ( ) y(t)=re{ae j e j( t) } realpartof complex exponential j = e e j t complex numbers s=a+jb j = s=re j conversion a = r cos( ) b = r sin( ) X= Ae j complex amplitude (constant) y(t)=re{xe j( t) } rotating phasor Euler s relations e j =cos( )+jsin( ) cos = ej + e j sin = ej e j j conjugate s*=a-jb=re -j e j0 = r = a + b e j / =i b = atan e j =- a quadrants!

7 complex numbers complex numbers s=a+jb j = s=re j conversion conjugate s*=a-jb=re -j a = r cos( ) b = r sin( ) r = a + b b = a tan a remember: e j0 = e j / =j e j =- ex. j =(e j0 ) j =e j(j)(0) =e -(0) =e 0 = ex. cos(j)? /j? s =3+j. jatan = 3. e 3 = 3 s =-+j e j j. atan =. e = 5 e j.678 s = 3 e j = =3+j s e j.678 = 5 = 5. cos cos j. 3 sin =-+j j. 5 sin Addition s =a +jb s +s =a +jb +a +jb s =a +jb =(a +a ) +j(b +b ) complex arithmetic Addition Subtraction Multiplication Division Powers Roots s =r e j s =r e j place tail of s at head of s convert to add convert back to connect origin to s

8 Addition - example Subtraction s =3+j s +s =a +a +j(b +b ) s =a +jb s -s =a +jb -(a +jb ) s =-+j = 3 - +j(+ ) s =a +jb =(a -a ) + j(b -b ) = +j3 s = 3 e j s = 3 cos(0.588)+j 3 sin(0.588)= 3+j s =r e j convert to add s = 5 e j.678 s s =r e j = 5 cos(.678)+j 5 sin(.678)= -+j convert back to s +s = +j3. = 3.e jatan 3 = rotate s 80 (-s ). 0.e j.49 place tail of -s at head of s connect origin to s Subtraction - example Multiplication s =a +jb s s =(a +jb )(a +jb ) s s =3+j -s =(a -a ) + j(b -b ) s =a +jb =a a +ja b +ja b +jb jb s =-+j =(3 - (-)) + j(- ) =a a +j b b + j(a b +a b ) =5 + j =a a -b b + j(a b +a b ) s e j s = 3 cos(0.588)+j 3 sin(0.588)= 3+j = 3 s = 5 e j.678 s = 5 cos(.678)+j 5 sin(.678)= -+j s =r e j s s =r e j r e j s =r e j =r r e j e j s -s = 5 +j. jatan = 5 5. e = j e. magnitudes multiply angles add =r r e j + x a x b =x a+b

9 =r e j (r e j ) Multiplication - example Division s s =a a -b b + j(a b +a b ) s =a +jb s /s =(a +jb )/(a +jb ) s =3+j = 3(-) -() + j(3()+()() ) s =a +jb s =-+j =(a +jb )(a -jb )/(a +jb )(a -jb ) = j(3-4 ) =(a +jb )(a -jb )/(a +b ) = -8 - j =s s * / s s = 3 e j s s =r r e j + s =r e j s /s =r e j / r e j s = 5 e j.678 = 3 5 e j( ) s =r e j = 65 e j3.66 =(r /r )e j e -j s = 65 cos(3.66)+ 65 s j sin(3.66) = -8 - j =(r /r )e j - divide magnitudes angles subtract x a /x b =x a-b Division - example s /s =s s * / s Powers =(a +jb )(a -jb )/(a +b ) s=a+jb s n =(a+jb) n s =3+j =(3+j ) (--j) / ( + Binomial expansion ) n =(3(-)-j3()+j(-)+()) / (5) n n k k n s =-+j = a ( jb) where = k k k= 0 =(-6-j3-j4+) / (5) =(-4-j7) / (5) s n =(re j ) n s = 3 e j s=re j s n =r n e jn s = 5 e j.678 s /s =(r /r )e j - 3 =( / 5 ) e j / ) e j =( 5 n(n )(n )K(n (k )) k! 3. =. e j = 5 cos(-.09)+j 5 sin(-.09) =-0.8-j.4 (x a ) n =x an magnitude raised to power n angle multiplied by n

10 Powers - example s =(+j) Roots s=+j s=a+jb s /n =(a+jb) /n??? / n 3 jatan. / n b b /n b /n b = a = + j + j + j. + j e a n a a 3 a =. s=re j = j e 5e. j = +K s=re j s /n =(re j ) /n s /n =r /n e j( /n+ k/n) k=0,,,,n- s n =r n e jn s =r e j s = 5 e j(0.464) s =5e j0.97 =5cos(0.97)+j5sin(0.97) =3+j4 n nth positive root of magnitude circle evenly divided by n starting at angle/n x = x / n Roots - example s=+j s /3 =(a+jb) /3 s /n =r /n e j( /n+ k/n) s /3 = 5 /3 e j(0.464/3+ k/3) =. j e. s /3 =.308 e j(0.464/3) =.308 e j(0.464/3+ /3) k=0,,,,n- k=0,, Complex Conversions s=a+jb s = a + b e j a tan b a s=re j s = rcos + jr sin Addition Complex Arithmetic ( a + jb ) + ( a + jb ) = ( a + a ) + j( b + b ) =.308 e j(0.464/3+ /3) s /3 =.308 e j0.55 =.9+j0.0 =.308 e j.49 =-0.8+j.09 =.308 e j4.343 =-0.47-j. Subtraction Multiplication Division ( a + jb ) ( a + jb ) = ( a a ) + j( b b ) r e j r e j = r r e j ( + ) r e j r e j = r e j ( ) r Powers ( re j ) n = r n e jn Roots z n = s = re j z = s / n = r / n j / n + k / n e ( ) k =,Kn