Signal Procssing First Lctur 5 Priodic Signals, Harmonics & im-varying Sinusoids READING ASSIGNMENS his Lctur: Chaptr 3, Sctions 3- and 3-3 Chaptr 3, Sctions 3-7 and 3-8 Nxt Lctur: Fourir Sris ANALYSIS Sctions 3-4, 3-5 and 3-6 /8/5 3, JH McClllan & RW Schar /8/5 3, JH McClllan & RW Schar 3 Problm Solving Sills LECURE OBJECIVES Math Formula Sum o Cosins Amp, Frq, Phas Rcordd Signals Spch Music No simpl ormula Plot & Stchs S( vrsus t Spctrum MALAB Numrical Computation Plotting list o numbrs Signals with HARMONIC Frquncis Add Sinusoids with = N = A + A cos(π t + ϕ ) = FREQUENCY can chang vs. IME Chirps: = cos(αt ) Introduc Spctrogram Visualization (spcgram.m) (plotspc.m) /8/5 3, JH McClllan & RW Schar 4 /8/5 3, JH McClllan & RW Schar 5
SPECRUM DIAGRAM Rcall Complx Amplitud vs. Frq X / 3 X = a 7 jπ / 3 7 jπ jπ / 4 4 jπ / jϕ X = A SPECRUM or PERIODIC? Narly Priodic in th Vowl Rgion Priod is (Approximatly) =.65 sc 5 5 = + 4 cos(π () t π / 3) + 8cos(π (5) t + π / ) (in Hz) /8/5 3, JH McClllan & RW Schar 6 /8/5 3, JH McClllan & RW Schar 7 PERIODIC SIGNALS Rpat vry scs Dinition Exampl: x ( = t + ) x ( = cos (3 Spch can b quasi-priodic /8/5 3, JH McClllan & RW Schar 8 = π 3 =? = π 3 Priod o Complx Exponntial j t = ω t + ) =? jω( t+ ) jω jω t Dinition: Priod is = = ω = π π π ω = = = ω j π = intgr /8/5 3, JH McClllan & RW Schar 9 =
Harmonic Signal Spctrum Priodic signal can only hav : = N = A + A cos(π = jϕ X = A = X + N j t j t { X + X } π π = t + ϕ ) /8/5 3, JH McClllan & RW Schar = Din FUNDAMENAL FREQ = A + A = N = cos(π ( ω = π = = undamntal Frquncy (largs = undamntal Priod (shorts /8/5 3, JH McClllan & RW Schar ) t + ϕ ) Harmonic Signal (3 Frqs) POP QUIZ: FUNDAMENAL 3rd 5th Hr s anothr spctrum: / 3 7 jπ 7 jπ jπ / 4 / 3 4 jπ / What is th undamntal rquncy? Hz 5 5 What is th undamntal rquncy? (in Hz) Hz? 5 Hz? /8/5 3, JH McClllan & RW Schar /8/5 3, JH McClllan & RW Schar 3
IRRAIONAL SPECRUM Harmonic Signal (3 Frqs) =. SPECIAL RELAIONSHIP to gt a PERIODIC SIGNAL /8/5 3, JH McClllan & RW Schar 4 /8/5 3, JH McClllan & RW Schar 5 NON-Harmonic Signal FREQUENCY ANALYSIS Now, a much HARDER problm Givn a rcording o a song, hav th computr writ th music NO Can a machin xtract rquncis? Ys, i w COMPUE th spctrum or During short intrvals /8/5 3, JH McClllan & RW Schar PERIODIC 6 /8/5 3, JH McClllan & RW Schar 7
im-varying FREQUENCIES Diagram SIMPLE ES SIGNAL Frquncy is th vrtical axis A-44 im is th horizontal axis C-major SCALE: stppd rquncis Frquncy is constant or ach not IDEAL /8/5 3, JH McClllan & RW Schar 8 /8/5 3, JH McClllan & RW Schar 9 R-ratd: ADULS ONLY SPECROGRAM ool MALAB unction is spcgram.m SP-First has plotspc.m & spctgr.m ANALYSIS program as as input & Producs spctrum valus X Bras into SHOR IME SEGMENS hn uss th FF (Fast Fourir ransorm) SPECROGRAM EXAMPLE wo Constant Frquncis: Bats cos( π (66) sin(π () /8/5 3, JH McClllan & RW Schar /8/5 3, JH McClllan & RW Schar
AM Radio Signal Sam as BEA Nots cos( π (66) sin(π () jπ (66) t jπ (66) t jπ () t j () t ( + ) ( ) π j jπ (67) t jπ (67) t jπ (648) t j (648) t ( + ) 4 π j π cos( π (67) t ) + cos(π (648) t + ) π SPECRUM o AM (Ba 4 complx xponntials in AM: j π j π / π / π / j / j 4 4 4 4 67 648 648 67 (in Hz) What is th undamntal rquncy? 648 Hz? 4 Hz? /8/5 3, JH McClllan & RW Schar /8/5 3, JH McClllan & RW Schar 3 SEPPED FREQUENCIES C-major SCALE: succssiv sinusoids Frquncy is constant or ach not SPECROGRAM o C-Scal Sinusoids ONLY IDEAL From SPECGRAM ANALYSIS PROGRAM ARIFACS at ransitions /8/5 3, JH McClllan & RW Schar 4 /8/5 3, JH McClllan & RW Schar 5
Spctrogram o LAB SONG Sinusoids ONLY Analysis Fram = 4ms ARIFACS at ransitions im-varying Frquncy Frquncy can chang vs. tim Continuously, not stppd FREQUENCY MODULAION (FM) = cos( π c t + v( ) VOICE CHIRP SIGNALS Linar Frquncy Modulation (LFM) /8/5 3, JH McClllan & RW Schar 6 /8/5 3, JH McClllan & RW Schar 7 Nw Signal: Linar FM Calld Chirp Signals (LFM) Quadratic phas = Acos( α t + π t + ϕ) Frq will chang LINEARLY vs. tim Exampl o Frquncy Modulation (FM) Din instantanous rquncy QUADRAIC /8/5 3, JH McClllan & RW Schar 8 INSANANEOUS FREQ Dinition = Acos( ψ ( ) ω i ( = ψ ( For Sinusoid: d dt d dt = Acos(π t + ϕ) ψ ( = π t + ϕ ω ( = i ψ ( = π Drivativ o th Angl Mas sns /8/5 3, JH McClllan & RW Schar 9
INSANANEOUS FREQ o th Chirp CHIRP SPECROGRAM Chirp Signals hav Quadratic phas Frq will chang LINEARLY vs. tim = ψ ( Acos( αt = αt + β t + ϕ) + β t + ϕ d i dt ω ( = ψ ( = αt + β /8/5 3, JH McClllan & RW Schar 3 /8/5 3, JH McClllan & RW Schar 3 CHIRP WAVEFORM OHER CHIRPS ψ( can b anything: = Acos( α cos( β + ϕ) d ω ( = ψ ( = αβ sin( β i dt ψ( could b spch or music: FM radio broadcast /8/5 3, JH McClllan & RW Schar 3 /8/5 3, JH McClllan & RW Schar 33
SINE-WAVE FREQUENCY MODULAION (FM) Loo at CD-ROM Dmos in Ch 3 /8/5 3, JH McClllan & RW Schar 34