Chapter 3. The Fourier Series

Transcription

1 Chpr 3 h Fourir Sris

2 Signls in h im nd Frquny Domin INC Signls nd Sysms Chpr 3 h Fourir Sris

3 Eponnil Funion r j ros jsin ) INC Signls nd Sysms Chpr 3 h Fourir Sris

4 Odd nd Evn Evn funion : Odd funion : ) ) ) ) If ) nd y) r vn or odd) : N N ) y ) d N ) y ) d If ) is vn nd y) is odd : N N ) y ) d INC Signls nd Sysms Chpr 3 h Fourir Sris

5 h Fourir Sris Fourir Sris : Priodi signls ould b rprsnd by sum of sinusoids or ompl ponnils. INC Signls nd Sysms Chpr 3 h Fourir Sris

6 h Fourir Sris Compl Eponnil Sris whr : j ), is h offiins of h Fourir Sris is h fundmnl frquny in rd/s) = / ) is h fundmnl priod is h hrmoni numbr =, ±, ±, ) INC Signls nd Sysms Chpr 3 h Fourir Sris

7 h Fourir Sris Compl Eponnil Sris ) j d,,,,... h n b ompud by ingrion ovr ny full priod. ) j d,,,,... INC Signls nd Sysms Chpr 3 h Fourir Sris

8 h Fourir Sris Compl Eponnil Sris ) j d,,,,... In s = ; ) d whr is h onsn or d omponn of ) In s = ± : h fundmnl omponn = ±n : h n h hrmoni omponn INC Signls nd Sysms Chpr 3 h Fourir Sris

9 h Fourir Sris Compl Eponnil Sris j ), ) j d,,,,... - If h ingrl dos no onvrg, CFS of h signl nno b found. - h signls r lwys rprsnd by linr ombinions of ompl sinusoids h fundmnl frquny nd i hrmonis INC Signls nd Sysms Chpr 3 h Fourir Sris

10 h Fourir Sris Compl Eponnil Sris ) j d,,,,... For ny rl signl, ); j, j INC Signls nd Sysms Chpr 3 h Fourir Sris

11 h Fourir Sris rigonomri C FS ) [ os ) b sin )], b )os ) d,,,... )sin ) d,,,... ) d INC Signls nd Sysms Chpr 3 h Fourir Sris

12 Chpr 3 h Fourir Sris INC Signls nd Sysms h Fourir Sris rigonomri Cosin-wih-phs form A, ) os ),,..., b A whn,,..., n whn,,...,, n b b

13 Chpr 3 h Fourir Sris INC Signls nd Sysms h offiin of FS Eponnil form rigonomri form,...,,, ) d j j,,..., ) )os d,,..., ) )sin d b d ) d ) b j ) ) jb ) jb A,,..., b A b n

14 Lin Spr ) os ).5os4 ) os8 ) 3.5,.56, , 4.5 6, INC Signls nd Sysms Chpr 3 h Fourir Sris

15 FS : Empl FS of h rngulr puls rin ) -/ - / d d ) INC Signls nd Sysms Chpr 3 h Fourir Sris

16 Chpr 3 h Fourir Sris INC Signls nd Sysms FS : Empl FS of h rngulr puls rin ) sin ) sin ) j j j d d j j j j j j j - / -/ )

17 FS : Empl h sin funion sin u) sin u ), u u sinu) INC Signls nd Sysms Chpr 3 h Fourir Sris

18 FS : Empl FS of h rngulr puls rin ) -/ - / Duy yl sin u) sin u) u sin ) sin ) sin ) INC Signls nd Sysms Chpr 3 h Fourir Sris

19 FS : Empl FS of h rngulr puls rin Duy yl = 5% ) -/4 /4 Duy yl =.5% ) -/6/6 INC Signls nd Sysms Chpr 3 h Fourir Sris

20 FS : Empl FS of h rngulr puls rin ) )os ) d os ) d os ) d.5.5 sin ) sin ) [sin )],,,....5 b.5 )sin ) d sin ) d sin ) d os ) [],.5.5 os ),,....5 INC Signls nd Sysms Chpr 3 h Fourir Sris

21 FS : Empl FS of h rngulr puls rin ) ) d 4 ) d.5 ) d N N ) sin )os ), odd INC Signls nd Sysms Chpr 3 h Fourir Sris

22 Gibbs Phnomnon N N ) sin )os ), odd N = 3 N = 9 ovrshoo 9% N = N = 45 INC Signls nd Sysms Chpr 3 h Fourir Sris

23 Prsvl s horm h vrg powr of signl n b luld by summing h squr of h mgniud of h Fourir offiins. Avrg powr P): P ) d P INC Signls nd Sysms Chpr 3 h Fourir Sris