2 T. or T. DSP First, 2/e. This Lecture: Lecture 7C Fourier Series Examples: Appendix C, Section C-2 Various Fourier Series
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1 DSP Firs, Lcur 7C Fourir Sris Empls: Common Priodic Signls READIG ASSIGMES his Lcur: Appndi C, Scion C- Vrious Fourir Sris Puls Wvs ringulr Wv Rcifid Sinusoids lso in Ch. 3, Sc. 3-5 Aug 6 3-6, JH McCllln & RW Schfr 3 LECURE OBJECIVES Us h Fourir Sris Ingrl d Driv Fourir Sris coffs for common priodic signls Drw spcrum from h Fourir Sris coffs is Compl Ampliud for -h Hrmonic Hrmonic Signl is Priodic F Sums of Hrmonic compl ponnils r Priodic signls PERIODFREQUECY of COMPLEX EXPOEIAL: F or F Aug 6 3-6, JH McCllln & RW Schfr Aug 6 3-6, JH McCllln & RW Schfr 5
2 Rcll FWRS sin Priod is Absolu vlu flips h ngiv lobs of sin wv Aug 6 3-6, JH McCllln & RW Schfr 6 FWRS Fourir Coffs: is funcion of Compl Ampliud for -h Hrmonic FWRS Fourir Ingrl { } sin d d d Full-Wv Rcifid Sin sin Priod : sin Aug 6 3-6, JH McCllln & RW Schfr 7 Spcrum from Fourir Sris Plo for Full-Wv Rcifid Sinusoid F nd F Dos no dpnd on h priod, DC vlu is Aug 6 3-6, JH McCllln & RW Schfr 8 Aug 6 3-6, JH McCllln & RW Schfr 9
3 Fourir Sris Synhsis HOW do you APPROXIMAE? d Us FIIE numbr of cofficins F * whn is rl Rconsruc From Fini umbr of Hrmonic Componns Full-Wv Rcifid Sinusoid sin i ms F Hz F F How clos is o sin? Aug 6 3-6, JH McCllln & RW Schfr Full-Wv Rcifid Sin { } is rl vlud cos cos... cos Plos for = nd =9 r shown n Eclln Approimion for =9 Aug 6 3-6, JH McCllln & RW Schfr... Aug 6 3-6, JH McCllln & RW Schfr Rconsruc From Fini umbr of Spcrum Componns Full-Wv Rcifid Sinusoid sin i Fms F Hz.6336 Aug 6 3-6, JH McCllln & RW Schfr 3
4 Fourir Sris Synhsis Fourir Sris Synhsis Aug 6 3-6, JH McCllln & RW Schfr PULSE WAVE SIGAL GEERAL FORM GEERAL FORM Dfind ovr on priod Dfind ovr on priod onzro DC vlu Aug 6 3-6, JH McCllln & RW Schfr 5 Puls Wv { } { } Gnrl PulsWv d d Gnrl PulsWv d sin Aug 6 3-6, JH McCllln & RW Schfr 6 Puls Wv { } = sinc { } PulsWv,...,, sin sin li Doubl chc h DC cofficin: d sin lim o, d Aug 6 3-6, JH McCllln & RW Schfr 7
5 PULSE WAVE SPECRA 6 Aug 6 3-6, JH McCllln & RW Schfr 8 PULSE WAVE SYHESIS wih firs 5 Hrmonics 5% duy-cycl Squr Wv sin sin,,,... hus, = whn is odd Phs is zro bcus is cnrd = diffrn from prvious cs PulsWv sring =, 3,,, Aug 6 3-6, JH McCllln & RW Schfr 9 ringulr Wv: im Domin Dfi Dfind ovr on priod id for onzro DC vlu 6 Aug 6 3-6, JH McCllln & RW Schfr Aug 6 3-6, JH McCllln & RW Schfr
6 ringulr Wv { } d ringulr Wv d ringulr Wv d d Aug 6 3-6, JH McCllln & RW Schfr us h indfini ingrl d ringulr Wv { } 3,..., Priod Ar DC Aug 6 3-6, JH McCllln & RW Schfr 3 3,..., ringulr Wv { } g { } S i 5 H i h Spcrum, ssuming 5 Hz is h fundmnl frquncy,... 3,,..., Aug 6 3-6, JH McCllln & RW Schfr ringulr Wv Synhsis 5 Hz ms F Aug 6 3-6, JH McCllln & RW Schfr 5
7 Full-Wv Rcifid Sin { } d Full Wv Rcifid Sin sin d Full-Wv Rcifid Sin : sin Priod d : Priod d d d d d Aug 6 3-6, JH McCllln & RW Schfr 6 Full-Wv Rcifid Sin { } Aug 6 3-6, JH McCllln & RW Schfr 7 Hlf-Wv Rcifid Sin Hlf-Wv Rcifid Sin Signl is posiiv hlf cycls of sin wv HWRS = Hlf-Wv Rciifid Sin Aug 6 3-6, JH McCllln & RW Schfr 8 Hlf-Wv Rcifid Sin { } d Hlf Wv Rcifid Sin sin d Hlf-Wv Rcifid Sin sin d d d d d d d Aug 6 3-6, JH McCllln & RW Schfr 9
8 Hlf-Wv Rcifid Sin { } odd? vn Aug 6 3-6, JH McCllln & RW Schfr 3 HWRS Synhsis Hlf-Wv Rcifid Sin { } Spcrum, ssuming 5 Hz is h fundmnl frquncy odd vn Aug 6 3-6, JH McCllln & RW Schfr 3 Fourir Sris Dmo F ms 5 Hz MALAB GUI: fsrisdmo Shows h convrgnc wih mor rms On of h dmos in: hp:dspfirs.gch.dumlb Aug 6 3-6, JH McCllln & RW Schfr 3 Aug 6 3-6, JH McCllln & RW Schfr 33
9 fsrisdmo GUI Aug 6 3-6, JH McCllln & RW Schfr 3
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