ELEG- Signal Processing and Communicaions Lecure 5 Periodic Signals, Harmonics & ime-varying Sinusoids READING ASSIGNMENS his Lecure: Chaper 3, Secions 3- and 3-3 Chaper 3, Secions 3-7 and 3-8 Lab sars sec., A, A rec. oday Nex Lecure: Fourier Series ANALYSIS Secions 3-, 3-5 and 3-6 ECE- Signal Processing Firs Problem Solving Sills LECURE OBJECIVES Mah Formula Sum o Cosines Amp, Freq, Phase Recorded Signals Speech Music No simple ormula Plo & Seches S versus Specrum MALAB Numerical Compuaion Ploing lis o numbers Signals wih HARMONIC Frequencies Add Sinusoids wih N x A A cosπ FREQUENCY can change vs. IME x cosα Chirps: Inroduce Specrogram Visualizaion specgram.m plospec.m ECE- Signal Processing Firs 3 ECE- Signal Processing Firs
SPECRUM DIAGRAM Recall Complex Ampliude vs. Freq X / 3 X a 7e jπ / 3 7e jπ jπ / e e jπ / j X A e 5 5 x cosπ π / 3 8cosπ 5 π / in Hz ECE- Signal Processing Firs 5 SPECRUM or PERIODIC? Nearly Periodic in he Vowel Region Period is Approximaely.65 sec ECE- Signal Processing Firs 6 PERIODIC SIGNALS Repea every secs Deiniion x x Example: x cos 3 Speech can be quasi-periodic? ECE- Signal Processing Firs 7 π 3 π 3 Period o Complex Exponenial x e x jω e e j ω jω x? jω Deiniion: Period is e ω π π π ω ω j e π ineger ECE- Signal Processing Firs 8
Harmonic Signal Specrum hereore, we can only have : N A cosπ x A X x X A e N j j { X e X e } π π j ECE- Signal Processing Firs 9 DEFINE FUNDAMENAL x A A N cosπ ω π undamenal Frequency undamenal Period ECE- Signal Processing Firs Harmonic Signal 3 Freqs POP QUIZ: FUNDAMENAL 3rd 5h Here s anoher specrum: / 3 7e jπ 7e jπ jπ / / 3 e e jπ / Wha is he undamenal requency? Hz 5 5 Wha is he undamenal requency? in Hz Hz? 5 Hz? ECE- Signal Processing Firs ECE- Signal Processing Firs 3
IRRAIONAL SPECRUM Harmonic Signal 3 Freqs. SPECIAL RELAIONSHIP o ge a PERIODIC SIGNAL ECE- Signal Processing Firs 3 ECE- Signal Processing Firs NON-Harmonic Signal FREQUENCY ANALYSIS Now, a much HARDER problem Given a recording o a song, have he compuer wrie he music NO Can a machine exrac requencies? Yes, i we COMPUE he specrum or x During shor inervals ECE- Signal Processing Firs PERIODIC 5 ECE- Signal Processing Firs 6
ime-varying FREQUENCIES Diagram Frequency is he verical axis A- ime is he horizonal axis SIMPLE ES SIGNAL C-major SCALE: sepped requencies Frequency is consan or each noe IDEAL ECE- Signal Processing Firs 7 ECE- Signal Processing Firs 8 SPECRUM ANALYSIS SPECROGRAM ool MALAB uncion is specgram.m ANALYSIS program aes x as inpu Produces specrum values X Breas x ino SHOR IME SEGMENS hen uses he FF Fas Fourier ransorm SPECROGRAM EXAMPLE wo Consan Frequencies: Beas cos π 66 cosπ ECE- Signal Processing Firs 9 ECE- Signal Processing Firs 5
AM Radio Signal Same as BEA Noes cos π 66 cosπ jπ 66 jπ 66 j j e e π e e jπ 67 jπ 67 jπ 68 j e e e e 68 π π cos π 67 cosπ 68 SPECRUM o AM Bea complex exponenials in AM: 67 68 Wha is he undamenal requency? 68 Hz? Hz? 68 67 in Hz ECE- Signal Processing Firs ECE- Signal Processing Firs SEPPED FREQUENCIES C-major SCALE: successive sinusoids Frequency is consan or each noe SPECROGRAM o C-Scale Sinusoids ONLY IDEAL From SPECGRAM ANALYSIS PROGRAM ARIFACS a ransiions ECE- Signal Processing Firs 3 ECE- Signal Processing Firs 6
Specrogram o Sinusoid Synhesis FUR ELISE SPECROGRAM Sinusoids ONLY Analysis Frame ms ARIFACS a ransiions ECE- Signal Processing Firs 5 ECE- Signal Processing Firs 6 ime-varying Frequency Frequency can change vs. ime Coninuously, no sepped FREQUENCY MODULAION FM x cosπ c v CHIRP SIGNALS Linear Frequency Modulaion LFM VOICE New Signal: Linear FM Called Chirp Signals LFM Quadraic phase x Acos α π Freq will change LINEARLY vs. ime Example o Frequency Modulaion FM Deine insananeous requency QUADRAIC ECE- Signal Processing Firs 7 ECE- Signal Processing Firs 8 7
8 ECE- Signal Processing Firs 9 INSANANEOUS FREQ Deiniion For Sinusoid: Derivaive o he Angle cos A x d d ω i Maes sense cos A x d d i π ω π π ECE- Signal Processing Firs 3 INSANANEOUS FREQ o he Chirp Chirp Signals have Quadraic phase Freq will change LINEARLY vs. ime β α β α A x cos β α ω d d i ECE- Signal Processing Firs 3 CHIRP SPECROGRAM ECE- Signal Processing Firs 3 CHIRP WAVEFORM
OHER CHIRPS SINE-WAVE FREQUENCY MODULAION FM can be anyhing: x Acos α cos β d ω αβ sin β i d could be speech or music: FM radio broadcas Loo a CD-ROM Demos in Ch 3 ECE- Signal Processing Firs 33 ECE- Signal Processing Firs 3 BIRD CHIRP SPECROGRAM ECE- Signal Processing Firs 35 9