Circuits and Systems I

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1 Circuis and Sysms I LECTURE #3 Th Spcrum, Priodic Signals, and h Tim-Varying Spcrum lions@pfl Prof. Dr. Volan Cvhr LIONS/Laboraory for Informaion and Infrnc Sysms

2 Licns Info for SPFirs Slids This wor rlasd undr a Craiv Commons Licns wih h following rms: Aribuion Th licnsor prmis ohrs o copy, disribu, display, and prform h wor. In rurn, licnss mus giv h original auhors crdi. Non-Commrcial Th licnsor prmis ohrs o copy, disribu, display, and prform h wor. In rurn, licnss may no us h wor for commrcial purposs unlss hy g h licnsor's prmission. Shar Ali Th licnsor prmis ohrs o disribu drivaiv wors only undr a licns idnical o h on ha govrns h licnsor's wor. Full Tx of h Licns This (hiddn pag should b p wih h prsnaion

3 Oulin - Today Today <> Scion Scion 3-7 Scion 3-8 Nx w <> Scion 3-4 Scion 3-5 Scion 3-6 Lab READ CSI Progrss Lvl:

4 Lcur Obcivs Sinusoids wih DIFFERENT frquncis SYNTHESIZE by Adding Sinusoids SPECTRUM Rprsnaion N 0 A cos( f 1 x( A ϕ Graphical Form shows DIFFERENT Frqs CSI Progrss Lvl:

5 FREQUENCY DIAGRAM A-440 Frquncy is h vrical axis Tim is h horizonal axis

6 Anohr FREQ. Diagram Plo Complx Ampliud vs. Frq 4 10 N 0 A cos( f 1 x( A ϕ / 3 7 / 3 7 / 4 / f (in Hz

7 Moivaion Synhsiz Complicad Signals Musical Nos Piano uss 3 srings for many nos Chords: play svral nos simulanously Human Spch Vowls hav dominan frquncis Applicaion: compur gnrad spch Can all signals b gnrad his way? Sum of sinusoids?

8 Fur Elis WAVEFORM Ba Nos

9 Spch Signal: BAT Narly Priodic in Vowl Rgion Priod is (Approximaly T sc

10 Eulr s Formula Rvrsd Solv for cosin (or sin sin( cos( ω ω ω sin( cos( ω ω ω sin( cos( ω ω ω cos( ω ω ω ( cos( 1 ω ω ω

11 INVERSE Eulr s Formula Solv for cosin (or sin cos( ω 1 ( ω ω sin( ω 1 ( ω ω

12 SPECTRUM Inrpraion Cosin sum of complx xponnials: Acos(7 A 7 A 7 On has a posiiv frquncy Th ohr has ngaiv frq. Ampliud of ach is half as big

13 SPECTRUM of SINE Sin sum of complx xponnials: Asin(7 1 A A A 1 Posiiv frq. has phas -0.5 Ngaiv frq. has phas 0.5 A

14 Ngaiv Frquncy Is ngaiv frquncy ral? Dopplr Radar provids an xampl Polic radar masurs spd by using h Dopplr shif principl L s assum 400Hz ß à 60 mph 400Hz mans owards h radar -400Hz mans away (opposi dircion Thin of a rain whisl

15 Graphical Spcrum EXAMPLE of SINE Asin(7 1 A A ( 1 A 0.5 ( 1 A AMPLITUDE, PHASE & FREQUENCY ar shown 7 ω

16 SPECTRUMSINUSOID 4 Add h spcrum componns: 10 / 3 7 / 3 7 / 4 / f (in Hz Wha is h formula for h signal x(?

17 Gahr (A,ω,φ informaion Frquncis: Ampliud & Phas -50 Hz -100 Hz 0 Hz 100 Hz 50 Hz 4 -/ 7 / /3 4 / No h conuga phas DC is anohr nam for zro-frq componn DC componn always has φ0 or (for ral x(

18 Add Spcrum Componns-1 Frquncis: -50 Hz -100 Hz 0 Hz 100 Hz 50 Hz Ampliud & Phas 4 -/ 7 / /3 4 / x( / 3 / (100 ( / 3 / (100 (50

19 4 Add Spcrum Componns- 10 / 3 7 / 3 7 / 4 / f (in Hz x( / 3 / (100 ( / 3 / (100 (50

20 Us Eulr s Formula o g REAL sinusoids: Simplify Componns x (50 / (50 / (100 / 3 (100 / ( A A A ω ϕ ω ϕ ϕ ω 1 1 cos(

21 Final Answr x( cos( (100 / 3 8cos( (50 / So, w g h gnral form: N 0 A cos( f 1 x( A ϕ

22 Summary: Gnral Form R z z z 1 1 } { f A X Frquncy ϕ { } R N f X X x 1 0 ( { } N f f X X X x ( N f A A x 1 0 cos( ( ϕ

23 Exampl: Synhic Vowl Sum of 5 Frquncy Componns

24 SPECTRUM of VOWEL No: Spcrum has 0.5X (xcp X DC Conugas in ngaiv frquncy

25 SPECTRUM of VOWEL (Polar Forma 0.5A φ

26 Vowl Wavform (sum of all 5 componns

27 Problm Solving Sills Mah Formula Sum of Cosins Amp, Frq, Phas Plo & Schs S( vrsus Spcrum Rcordd Signals Spch Music No simpl formula MATLAB Numrical Compuaion Ploing lis of numbrs

28 Lcur Obcivs Signals wih HARMONIC Frquncis Add Sinusoids wih f f 0 N A cos( f x( A ϕ FREQUENCY can chang vs. TIME Chirps: x( cos(α Inroduc Spcrogram Visualizaion (spcgram.m (plospc.m!

29 Spcrum Diagram 4 Rcall Complx Ampliud vs. Frq 10 X 1 / 3 7 / 3 7 / 4 / 1 X A ϕ X a f (in Hz x( cos( (100 / 3 8cos( (50 /

30 Spcrum for Priodic Signals? Narly Priodic in h Vowl Rgion Priod is (Approximaly T sc

31 Priodic Signals Rpa vry T scs Dfiniion x ( x( T Exampl: x ( cos (3 T? T 3 T 3 Spch can b quasi-priodic

32 Priod of Complx Exponnials Dfiniion: Priod is T ingr T ω ω (? ( ( ( x T x x ω 1 T T ω ω 1 T T 0 ω ω

33 { } N f f N X X X x A X f A A x ( cos( ( ϕ ϕ Harmonic Signal Spcrum 0 can only hav : signal Priodic f f T f 1 0

34 Dfin Fundamnal Frquncy T f Priod (shors fundamnal Frquncy (largs fundamnal ( cos( ( T f f f f f A A x N ω ϕ

35 Harmonic Signal (3 Frqs 3rd 5h Wha is h fundamnal frquncy? 10 Hz

36 POP QUIZ: Fundamnal Frq. Hr s anohr spcrum: 4 10 / 3 7 / 3 7 / 4 / f (in Hz Wha is h fundamnal frquncy? 100 Hz? 50 Hz?

37 IRRATIONAL SPECTRUM SPECIAL RELATIONSHIP o g a PERIODIC SIGNAL

38 Harmonic Signal (3 Frqs T0.1

39 NON-Harmonic Signal NOT PERIODIC

40 Frquncy Analysis Now, a much HARDER problm Givn a rcording of a song, hav h compur wri h music Can a machin xrac frquncis? Ys, if w COMPUTE h spcrum for x( During shor inrvals

41 Tim-Varying FREQUENCIES Diagram A-440 Frquncy is h vrical axis Tim is h horizonal axis

42 A Simpl Ts Signal C-maor SCALE: sppd frquncis Frquncy is consan for ach no IDEAL

43 R-rad: ADULTS ONLY SPECTROGRAM Tool MATLAB funcion is spcgram.m SP-Firs has plospc.m & spcgr.m ANALYSIS program Tas x( as inpu & Producs spcrum valus X Bras x( ino SHORT TIME SEGMENTS Thn uss h FFT (Fas Fourir Transform

44 R-rad: ADULTS ONLY SPECTROGRAM Tool MATLAB funcion is spcgram.m SP-Firs has plospc.m & spcgr.m ANALYSIS program Tas x( as inpu & CSI Progrss Lvl: Producs spcrum valus X Bras x( ino SHORT TIME SEGMENTS Thn uss h FFT (Fas Fourir Transform

45 Spcrogram Exampl Two Consan Frquncis: Bas cos( (660 sin( (1

46 AM Radio Signal Sam as BEAT Nos cos( (660 sin( (1 ( (660 (660 ( (1 (1 1 1 ( (67 (67 (648 ( cos( (67 1 cos( (648

47 Spcrum of AM (Ba 4 complx xponnials in AM: 1 / 4 1 / 4 1 / 4 1 / f (in Hz Wha is h fundamnal frquncy? 648 Hz? 4 Hz?

48 Sppd Frquncis C-maor SCALE: succssiv sinusoids Frquncy is consan for ach no IDEAL

49 Spcrogram of C-Scal Sinusoids ONLY From SPECGRAM ANALYSIS PROGRAM ARTIFACTS a Transiions

50 Spcrogram of LAB SONG Sinusoids ONLY Analysis Fram 40ms ARTIFACTS a Transiions

51 Tim-Varying Frquncy Frquncy can chang vs. im Coninuously, no sppd FREQUENCY MODULATION (FM x( cos( f v( c VOICE CHIRP SIGNALS Linar Frquncy Modulaion (LFM

52 Nw Signal: Linar FM Calld Chirp Signals (LFM Quadraic phas QUADRATIC x( Acos( α f 0 ϕ Frq will chang LINEARLY vs. im Exampl of Frquncy Modulaion (FM Dfin insananous frquncy

53 Insananous Frquncy Dfiniion For Sinusoid: Drivaiv of h Angl ( ( ( cos( ( A x d d ω i ψ ψ Mas sns ( ( ( cos( ( f f f A x d d i ψ ω ϕ ψ ϕ

54 Insananous Frquncy of h Chirp Chirp Signals hav Quadraic phas Frq will chang LINEARLY vs. im ϕ β α ψ ϕ β α A x ( cos( ( β α ψ ω d d i ( (

55 Chirp Spcrogram

56 Chirp Wavform

57 OTHER CHIRPS ψ( can b anyhing: x( Acos( α cos( β ϕ ψ( could b spch or music: FM radio broadcas ω i d d ( ψ ( αβ sin( β

58 Sin-Wav Frquncy Modulaion (FM Loo a CD-ROM Dmos in Ch 3

59 Nx w <> Scion 3-4 Scion 3-5 Scion 3-6 Lab

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