s=a+jb s=% x(t)=sin(&t)

Transcription

1 Linearity input Ax (t)+bx (t) Time invariance x(t-$) x(t) Characteristic Functions Linear Time Invariant Systems Linear Time Invariant System scaling & superposition e st complex exponentials y(t) output Ay (t)+by (t) y(t-$) s% s±j& s-% ±j& x(t)sin(&t) e st e -%t e ±j&t e -%±j&t cos" ej" + e #j" Complex Exponential Signals Linear Time Invariant System exponential decay sinusoids exponential sinusoids characteristic functions of LTI systems 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() "/6 " /4 " /3 " / " 3"/ " cosine even cos(-)-cos() # y cos() # Relations: &"f T:period (sec/cycle) Continuous sinusoids T /f "/& (t) 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) y(t)a sin(&t+') y(t)a sin("ft+') sec/cycle / (cycle/sec) (" rad/cycle)/(rad/sec)

2 Continuous sinusoids (t) Sampled Continuous Sinusoid Phase shift In: x(t)sin(t) Out: y(t).sin(t-"/3) 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) x() y("/3) Continuous Sinusoid [ ] sin(" # # n) [ ] sin( " # n) y n y n y(t) sin("t) Sample rate: t nt s Discrete Sinusoid T s sec y[n]cos(*pi*n/) TextEnd sampled continuous sinusoids n - samples n y[n] Discrete sinusoids [n] n,, 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 Nk/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) What is the period N? y[n]a cos("fn+') frequency: f3/6 cycles/sample N:period (samples/repeating cycle [integer]) Smallest integer N such that y[n]y[n+n] Find an integer k so Nk/f is also an integer f3/6 let k3 N(3)*6/36-3 n - samples f k/n (rational number -> k/n is ratio of integers) y[n]cos(*pi*3/6*n) TextEnd period of discrete sinusoids

3 y[ n] sin(" # 3 # n) f 3 cycles sample y[ n] y[n + N] N?? samples N: integer Period of discrete sinusoids: ex. discrete function y[ n] sin(" # 3 # n) frequency: period: N?? samples y[ n] y[n + N] f " N k Period of discrete sinusoids: ex. f 3 cycles sec N, k:integers 3 " N k N " f /3 integer N k 3 samples cycle ratio of integers rational number periodic N samples, k3 cycles Aperiodic discrete sinusoids continuous function ( ) sin(" # # t) y t T sec sample t nt s periodic T s sec ysin(*pi*sqrt()/*n) TextEnd irrational frequency arbitrary continuous signal y(t+()y(t) After what interval does the signal repeat itself? Periodicity T:period (sec/cycle) discrete function [ ] sin(" # y n period? [ ] y[n + N] y n N?? samples (integer) # n) time (sec) f f " N k N,k: integers N k not a ratio of integers irrational number sampled discrete sinusoid aperiodic

4 Ex: Period of sum of sinusoids T. seconds, T.7 seconds seconds to complete cycles T/ seconds /s, /s, 3/s 4/s, 8/s, /s, 6/s, /s, 4/s, 8/s, 3/s, 36/s, 4/s, 44/s, 48/s, /s, 6/s, 6/s cycles Least common multiple /*k3/4*l k/l/4 seconds to complete cycles T3/4 seconds 3/4s, 6/4s, /s. 3/s, 4/s, 6/s rational number 4 cycles - Tsum*T/3 seconds Tsum4*T3/4*43 seconds time (sec) Tsum? seconds Tsum3 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 & ((& -& )/T)t +& integrate (& -& )/T t +& t + C time varying argument t & & ( & y chirp (t)a sin((& -& ) /(T) t +& t + ') Representations of a sinusoid y(t)a cos(&t+') [ ] e j#t + e $ j#t y(t) Ae j" y(t)re{ae j' e j(&t) } X Ae j' complex amplitude (constant) y(t)re{xe j(&t) } Euler s relations e j cos()+jsin() trig function complex conjugates real part of complex exponential rotating phasor cos" ej" + e #j" sin" ej" # e #j" j e j (") e j (#t+$) e j$ e j#t

5 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) Vector representation (graphical) Properties of exponentials re x e y re x+y (re x ) n r n e nx n x x / n x x" Complex Exponentials Amplitude modulation (multiply two sinusoids of different frequencies) Acos(& t) B cos(& t+') C(cos(& 3 t +' )+ cos(& 4 t +' )) cos. cos " cos. cos. cos " sin. sin " cos. cos " cos. sin. sin " cos cos sin sin. cos ". sin " cos s cos d sin s sin d. cos ". sin "... A B cos. cos "s cos. cos "d sin. sin "s sin. sin "d. cos s " sin d " or cos. cos " e j.. e j. e j. " e j. " cos complex conj mult. exponentials... 4 A B exp j. " " exp j. " " exp j. " " exp j. " "... 4 A B. cos " ". cos " ". cos " " cos " " *sum formula for sin & cos trig id *product formula for sin & cos trig id *sum formula for sin & cos trig id cos complex conj Representations of a sinusoid complex numbers y(t)a cos(&t+') [ ] e j#t + e $ j#t y(t) Ae j" y(t)re{ae j' e j(&t) } trig function complex conjugates real part of complex exponential e j (") e j (#t+$) e j$ e j#t sre j 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-jbre -j r a + b # " atan% b& ( $ a' quadrants e j e j"/ i e j" -

6 complex numbers complex numbers sre j conjugate s*a-jbre -j j " conversion a r cos(" ) b r sin(" ) r a + b # " a tan b & % ( $ a ' remember: e j e j"/ j e j" - ex. j (e j ) j e j(j)() e -() e ex. cos(j)? /j? s 3+j 3. e s -+j. j atan 3 3 e j.88 j. atan. e e j.678 s 3 e j.88 3+j s e j cos.88 j. 3. sin.88. cos.678 j.. sin j 3 Addition complex arithmetic Addition Subtraction Multiplication Division Powers Roots s a +jb s +s a +jb + a +jb s a +jb (a + a ) +j(b +b ) s r e j s r e j convert to add convert back to place tail of s at head of s connect origin to s

7 Addition - example s 3+j s -+j s 3 e j.88 s e j.678 s +s a + a +j(b +b ) 3 - +j(+ ) +j3 s 3 cos(.88)+j 3 sin(.88) 3+j s cos(.678)+j sin(.678) -+j s +s +j3 3. j e. atan 3. e j..49 Subtraction s a +jb s -s a +jb - (a +jb ) s a +jb (a - a ) + j(b -b ) s r e j s r e j rotate s 8 (-s ) place tail of -s at head of s convert to add convert back to connect origin to s Subtraction - example s s -s (a - a ) + j(b -b ) 3+j s -+j (3 - (-)) + j(- ) + j s e j.88 s 3 3 cos(.88)+j 3 sin(.88) 3+j s e j.678 s cos(.678)+j sin(.678) -+j s -s +j j. atan. e 6. e j..97 Multiplication s a +jb s s (a +jb ) s a +jb (a +jb ) a a + ja b + ja b + jb jb a a + j b b + j(a b +a b ) a a -b b + j(a b +a b ) s r e j s r e j magnitudes multiply angles add s s r e j r e j r r e j e j r r e j+ x a x b x a+b

8 Multiplication - example s s a a -b b + j(a b +a b ) s 3+j 3(-) -() + j(3()+()() ) s -+j s 3 e j.88 s e j j(3-4 ) -8 - j s s r r e j+ 3 e j( ) 6 e j3.66 s s 6 cos(3.66)+ 6 j sin(3.66) -8 - j Division s a +jb s a +jb s r e j s r e j s /s (a +jb ) / (a +jb ) (a +jb ) (a -jb ) / (a +jb ) (a -jb ) (a +jb ) (a -jb ) / (a +b ) s s * / s s /s r e j / r e j r e j (r e j ) # (r /r ) e j e -j (r /r ) e j- divide magnitudes angles subtract x a /x b x a-b Division - example s 3+j s -+j s 3 e j.88 s e j.678 s /s s s * / s (a +jb ) (a -jb ) / (a +b ) (3+j ) (--j) / ( + ) (3(-)-j3()+j(-)+()) / () (-6-j3-j4+) / () (-4-j7) / () s /s (r /r ) e j- Powers sre j s n (a+jb) n n " n ( $ % ' a n)k ( jb) k " where n $ % ' # k& # k& k s n (re j ) n s n r n e jn Binomial expansion n(n ()(n ( )K(n ( ( k ( )) k 3 ( / ) e j.88# ( / ) ej.88# e j cos(-.9)+j sin(-.9) -.8-j.4 (x a ) n x an magnitude raised to power n angle multiplied by n

9 Powers - example s+j sre j. e. e j j atan. e j..464 s (+j) Roots sre j s /n (a+jb) /n " a/ n + j b % $ ' # a& / n s /n (re j ) /n s /n r /n e j(/n+"k/n) + j b n a + " /n % $ # & ' j b " % " $ # a ' + /n % $ & # 3 & ' j b 3 " % $ # a ' +K & k,,,,n-??? s n r n e jn s r e j s e j(.464) s e j.97 cos(.97)+jsin(.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. e j..464 s /3 (a+jb) /3 s /n r /n e j(/n+"k/n) s /3 /3 e j(.464/3+"k/3) s /3.38 e j(.464/3).38 e j(.464/3+"/3).38 e j(.464/3+"/3) k,,,,n- k,, s /3.38 e j..9+j..38 e j j.9.38 e j j. s a + b e j"a tan ( b a) sre j s rcos" + jrsin" Addition Subtraction Multiplication Division Powers Complex Conversions Complex Arithmetic ( a + jb ) + ( a + jb ) ( a + a ) + j( b + b ) ( 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 ( re j" ) n r n e jn" Roots z n s re j" z s / n r / n e j " / n +#k / n ( ) k,kn "