y[ n] = sin(2" # 3 # n) 50

Transcription

1 Period of a Discrete Siusoid y[ ] si( ) 5 T5 samples y[ ] y[ + 5] si() si() [ ] si( 3 ) 5 y[ ] y[ + T] T?? samples [iteger] 5/3 iteger y irratioal frequecy ysi(pisqrt()/5) - - TextEd si( t) T sec cotiuous fuctio periodic - - y[ ] si( 3 ) 5 y[ ] y[ + T] T?? samples si() si(k) k, periodic T5 samples, k3 cycles 3 k 5 k 5 samples 3 cycle Ratio of itegers ratioal umber [ ] si( y[ ] y[ + T] y time (sec) Ts/5 sec si() si(k) 5 ) T?? samples k, Equiv. discrete siusoid ot periodic k 5 k 5 irratioal umber

2 Period of Sum of Siusoids T secods, T.75 secods - y( t) y( t + T) time (sec) Tsum3 secods secods to complete cycles T/5 secods /5s, /5s, 3/5s /s, 8/s, /s, 6/s, /s, /s, 8/s, 3/s, 36/s, /s, /s, 8/s, 5/s, 56/s, 6/s 5 cycles Least commo multiple Tsum5T5/53 secods /5k3/l k/l5/ Tsum3 secods secods to complete cycles T3/ secods 3/s, 6/s, 5/s. 3/s, 5/s, 6/s ratioal umber cycles TsumT3/3 secods cartesia cartesia sa+jb s a + b e ja ta ( b a) sre j s rcos + jrsi Additio Subtractio Multiplicatio Divisio Powers Roots cartesia cartesia Complex Coversios 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 rr e j ( + ) r e j r e j r e j ( ) r ( re j ) r e j z s re j z s / r / e j / +k / k,k

3 cartesia cartesia Additio Subtractio Multiplicatio Divisio Powers Roots 3+ j 3 + e ja ta ( ) 3 5e j.97 cartesia cartesia Complex Coversios Complex Arithmetic e j 3 cos 3 + jsi 3 + j 3 j) + ( 3+ j) ( + j6) j) ( 3+ j) ( j) 5e j 3 6e j 5 6e j ( 3 + ) j 3e 7 e j 5e j ( 5 )e j ( ) j e 3e j e j 3 j 3 7e z 3 6 6e j z 6 / 3 e j( / 3+k / 3) e j( k / 3) e j( / 3) e j( / 3) j ft + Re{ Ae } Acos kf t + k Re Ae j e j ft Re Xe j ft Sum multiple cosies same frequecy Ex. Ae j e j ft +e j ft e X j ft +e j ft A k cos( ft + k ) Re A k e ft + k Re A k e k eft k Represetatios of Siusoids k ' Re A k e k ) e ft ( k cos( t 6 ) + cos( t + 3) 3cos t + Re 3e j e j t e j 6 e j t + e 3 e Re 3e j e j 6 + e 3 ' - j t + ) e., ( / Re 5.3e j.55 t e 5.3cos( t +.55) j t ' ( ) k multiply cosies of differet frequecy A cos ( t) A cos ( t + ) A e j t + e j t ' e j ( t + ) + e j ( t + ) ' ) A ) ( ( A A A A A A e j t e j ( t + ) + e j t e j ( t + ) + e j t e j ( t + ) + e j t e j ( t + ) e j ( t + t + ) + e j ( t t + ) + e j ( t t + ) + e j ( t + t + ) ( ) ( cos (( + )t + ) + cos (( )t + )) Composite sigals (waveform sythesis) x(t) A cos kf t + k k ) e j kf t X decompose a periodic sigal x(t) ito a sum of a series of siusoids - the Fourier series. Note: The sum of periodic fuctios is periodic. ex. 8 k odd k ' k eve k f 5Hz

4 Composite sigals (waveform sythesis) x(t) A cos kf t + k k k 8 k odd k ' k eve x(t) 5cos( 5t + ) ) e j kf t ( k + 8 k e j k odd k eve X Composite sigals (waveform sythesis) x(t) A cos kf t + k k 8 k e j k odd k eve k3 ) e j kf t X x(t) 5cos( 5t + ) +.9cos( 75t + ) k Composite sigals (waveform sythesis) x(t) A cos kf t + k k 8 k e j k odd k eve k5 ) e j kf t X x(t) 5cos( 5t + ) +.9cos( 75t + ) +.3cos( 5t + ) k X 5 spectrum 53e j 53e j e j.5e j e j.6e j f x(t) 5cos 5t + +.9cos( 75t + ) +.3 cos( 5t + ) +...

5 For a give sigal, how do we fid for each k? Fourier Aalysis x(t) A cos kf t + k where X T T k T T ) e j kf t X x(t)dt x(t)e j kt T dt A k e j k k f :fudametal frequecy T / f x(t) t t < T x(t) A cos kf t + k X T T k T x(t)dt tdt T T ) e j kf t X t T Mathematica: atheaadd math atheamath I[]:/TItegrate[t,{t,,T}] Out[]:T/ T T T k.. t t. T T x(t)e j kt T dt x(t) t t < T x(t) A cos kf t + k X T T T T T j T k te j kt T dt ( jk +) e jk T k k e jk ( e j ) k ( jk +) T k k k k k + T k T k j T k T k e j ) e j kf t X k Mathematica: I[]: /TItegrate[tExp[-IPikt/T],{t,,T}] ( I) k Pi -((- + E - ( I) k Pi) T) Out[] ( I) k Pi E k Pi I[3]: Simplify[,Elemet[k,Itegers]] I T Out[3] ---- k Pi j T k T k e j T e jk e jk k T te j kt T dt

6 x(t) t t < T x(t) A cos kf t + k X T k T k e j x(t) T + T cos kf t + k k ) e j kf t X T / f x(t) T + T cos ( f t + ) + T cos ( f t + ) +K k f :fudametal frequecy x(t) t t < T x(t) A cos kf t + k X T k T k e j x(t) T + T cos kf t + k k ) e j kf t X x(t) T + T cos ( f t + ) + T cos ( f t + ) +K k. t t. f 5Hz T / f. 7 terms. t t. Defied betwee <t<. Periodic with period. :Square Wave :Square Wave x(t) t < T T t < T z t. T T e j kt T dt + e j kt T dt T T T T X dt + dt T T T T t. I[]:/TItegrate[Exp[-IPikt/T],{t,,T/}]+ /TItegrate[- Exp[-IPikt/T],{t,T/,T}] -I k Pi I k Pi -I ( - E ) I (- + E ) Out[] k Pi ( I) k Pi E k Pi I[]:/TItegrate[,{t,,T/}]+ /TItegrate[-,{t,T/,T}] Out[]: X I[3]: Simplify[,Elemet[k,Itegers]] k -I (- + (-) ) Out[7] k Pi j (+ ( ) )k k j k odd k ' k eve j () k j k j (+) k j () k

7 x(t) t < T T t < T X j k odd k ' k eve k e j k odd ' k eve x(t) A cos kf t + k k ) e j kf t X k. x(t) cos f ( t ) + cos 3 f 3 ( t ) +K.5 z t t.